Knot Polynomials: Myths and Reality
This article provides an overview of relative strengths of polynomial invariants of knots and links, such as the Alexander, Jones, Homflypt, Kaufman two-variable polynomial, and Khovanov polynomial.
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arXiv subjects
Slavik Jablan
Publications and source records attributed to Slavik Jablan.
Tutte and Jones Polynomial of Link Families
This article contains general formulas for Tutte and Jones polynomial for families of knots and links given in Conway notation.
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Tutte and Jones polynomials of link families
This article contains general formulas for Tutte and Jones polynomials for families of knots and links given in Conway notation and "portraits of families"-- plots of zeroes of their corresponding Jones polynomials.
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Hyperbolic Volume of Link Families
For families of knots and links given in Conway notation we compute lower maximal and upper minimal bound of hyperbolic volume by using source links and augmented links.
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Adequacy of Link Families
Using computer calculations and working with representatives of pretzel tangles we established general adequacy criteria for different classes of knots and links. Based on adequate graphs obtained from all Kauffman states of an alternating link we defined a new numerical invariant: adequacy number, and computed adequacy polynomial which is the invariant of alternating link families. Adequacy polynomial distinguishes (up to mutation) all families of alternating knots and links whose generating link has at most $n=12$ crossings.
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Unlinking Number and Unlinking Gap
Computing unlinking number is usually very difficult and complex problem, therefore we define BJ-unlinking number and recall Bernhard-Jablan conjecture stating that the classical unknotting/unlinking number is equal to the BJ-unlinking number. We compute BJ-unlinking number for various families of knots and links for which the unlinking number is unknown. Furthermore, we define BJ-unlinking gap and construct examples of links with arbitrarily large BJ-unlinking gap. Experimental results for BJ-unlinking gap of rational links up to 16 crossings, and all alternating links up to 12 crossings are obtained using programs LinKnot and K2K. Moreover, we propose families of rational links with arbitrarily large BJ-unlinking gap and polyhedral links with constant non-trivial BJ-unlinking gap. Computational results suggest existence of families of non-alternating links with arbitrarily large BJ-unlinking gap.
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Braid Family Representatives
After defining reduced minimum braid word and criteria for a braid family representative, different braid family representatives are derived, and a correspondence between them and families of knots and links given in Conway notation is established.
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