# K11n13

 (Knotscape image) See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n13 at Knotilus!

### Knot presentations

 Planar diagram presentation X4251 X8394 X10,6,11,5 X16,8,17,7 X2,9,3,10 X11,19,12,18 X13,21,14,20 X6,16,7,15 X17,1,18,22 X19,13,20,12 X21,15,22,14 Gauss code 1, -5, 2, -1, 3, -8, 4, -2, 5, -3, -6, 10, -7, 11, 8, -4, -9, 6, -10, 7, -11, 9 Dowker-Thistlethwaite code 4 8 10 16 2 -18 -20 6 -22 -12 -14

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 3 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n13/ThurstonBennequinNumber Hyperbolic Volume 6.56631 A-Polynomial See Data:K11n13/A-polynomial

### Four dimensional invariants

 Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $4$ Rasmussen s-Invariant -6

### Polynomial invariants

 Alexander polynomial $-t^4+3 t^3-2 t^2+t-1+ t^{-1} -2 t^{-2} +3 t^{-3} - t^{-4}$ Conway polynomial $-z^8-5 z^6-4 z^4+4 z^2+1$ 2nd Alexander ideal (db, data sources) $\{1\}$ Determinant and Signature { 15, 6 } Jones polynomial $-q^{10}+q^9-q^8+2 q^7-2 q^6+2 q^5-2 q^4+2 q^3-q^2+q$ HOMFLY-PT polynomial (db, data sources) $-z^8 a^{-6} +z^6 a^{-4} -7 z^6 a^{-6} +z^6 a^{-8} +6 z^4 a^{-4} -16 z^4 a^{-6} +6 z^4 a^{-8} +10 z^2 a^{-4} -15 z^2 a^{-6} +10 z^2 a^{-8} -z^2 a^{-10} +4 a^{-4} -6 a^{-6} +5 a^{-8} -2 a^{-10}$ Kauffman polynomial (db, data sources) $z^9 a^{-5} +z^9 a^{-7} +z^8 a^{-4} +3 z^8 a^{-6} +2 z^8 a^{-8} -6 z^7 a^{-5} -5 z^7 a^{-7} +z^7 a^{-9} -7 z^6 a^{-4} -19 z^6 a^{-6} -12 z^6 a^{-8} +10 z^5 a^{-5} +6 z^5 a^{-7} -4 z^5 a^{-9} +16 z^4 a^{-4} +37 z^4 a^{-6} +23 z^4 a^{-8} +2 z^4 a^{-10} -4 z^3 a^{-5} -2 z^3 a^{-7} +3 z^3 a^{-9} +z^3 a^{-11} -14 z^2 a^{-4} -26 z^2 a^{-6} -19 z^2 a^{-8} -6 z^2 a^{-10} +z^2 a^{-12} +z a^{-7} -z a^{-9} -z a^{-11} +z a^{-13} +4 a^{-4} +6 a^{-6} +5 a^{-8} +2 a^{-10}$ The A2 invariant $q^{-4} + q^{-6} + q^{-8} + q^{-10} - q^{-16} - q^{-20} + q^{-22} + q^{-24} + q^{-26} - q^{-30} - q^{-34}$ The G2 invariant Data:K11n13/QuantumInvariant/G2/1,0

### "Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {9_2,}

### Vassiliev invariants

 V2 and V3: (4, 10)
V2,1 through V6,9:
 V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $16$ $80$ $128$ $\frac{1448}{3}$ $\frac{280}{3}$ $1280$ $\frac{8960}{3}$ $\frac{1760}{3}$ $464$ $\frac{2048}{3}$ $3200$ $\frac{23168}{3}$ $\frac{4480}{3}$ $\frac{279542}{15}$ $\frac{2392}{15}$ $\frac{371768}{45}$ $\frac{1306}{9}$ $\frac{16742}{15}$

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.

### Khovanov Homology

The coefficients of the monomials $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$6 is the signature of K11n13. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-2-101234567χ
21         1-1
19          0
17       11 0
15      1   1
13     11   0
11    11    0
9   11     0
7  11      0
5 12       1
3          0
11         1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=5$ $i=7$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=6$ ${\mathbb Z}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory. See A Sample KnotTheory Session.