# K11n1

 (Knotscape image) See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n1 at Knotilus!

 Knot K11n1. A graph, knot K11n1. A part of a knot and a part of a graph.

### Knot presentations

 Planar diagram presentation X4251 X8394 X10,6,11,5 X14,7,15,8 X2,9,3,10 X11,16,12,17 X13,20,14,21 X6,15,7,16 X17,22,18,1 X19,12,20,13 X21,18,22,19 Gauss code 1, -5, 2, -1, 3, -8, 4, -2, 5, -3, -6, 10, -7, -4, 8, 6, -9, 11, -10, 7, -11, 9 Dowker-Thistlethwaite code 4 8 10 14 2 -16 -20 6 -22 -12 -18
A Braid Representative

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n1/ThurstonBennequinNumber Hyperbolic Volume 9.70048 A-Polynomial See Data:K11n1/A-polynomial

### Four dimensional invariants

 Smooth 4 genus Missing Topological 4 genus Missing Concordance genus ${\displaystyle 2}$ Rasmussen s-Invariant 2

### Polynomial invariants

 Alexander polynomial ${\displaystyle -t^{2}+7t-11+7t^{-1}-t^{-2}}$ Conway polynomial ${\displaystyle -z^{4}+3z^{2}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{1\}}$ Determinant and Signature { 27, -2 } Jones polynomial ${\displaystyle q^{-1}-q^{-2}+3q^{-3}-4q^{-4}+4q^{-5}-4q^{-6}+4q^{-7}-3q^{-8}+2q^{-9}-q^{-10}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle -a^{10}+2z^{2}a^{8}+2a^{8}-z^{4}a^{6}-2z^{2}a^{6}-2a^{6}+2z^{2}a^{4}+a^{4}+z^{2}a^{2}+a^{2}}$ Kauffman polynomial (db, data sources) ${\displaystyle z^{7}a^{11}-5z^{5}a^{11}+7z^{3}a^{11}-3za^{11}+2z^{8}a^{10}-10z^{6}a^{10}+14z^{4}a^{10}-7z^{2}a^{10}+a^{10}+z^{9}a^{9}-2z^{7}a^{9}-7z^{5}a^{9}+13z^{3}a^{9}-4za^{9}+4z^{8}a^{8}-19z^{6}a^{8}+25z^{4}a^{8}-12z^{2}a^{8}+2a^{8}+z^{9}a^{7}-2z^{7}a^{7}-5z^{5}a^{7}+8z^{3}a^{7}-2za^{7}+2z^{8}a^{6}-9z^{6}a^{6}+13z^{4}a^{6}-9z^{2}a^{6}+2a^{6}+z^{7}a^{5}-3z^{5}a^{5}+3z^{3}a^{5}-za^{5}+2z^{4}a^{4}-3z^{2}a^{4}+a^{4}+z^{3}a^{3}+z^{2}a^{2}-a^{2}}$ The A2 invariant Data:K11n1/QuantumInvariant/A2/1,0 The G2 invariant Data:K11n1/QuantumInvariant/G2/1,0

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

Same Alexander/Conway Polynomial: {9_48,}

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

### Vassiliev invariants

 V2 and V3: (3, -7)
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 ${\displaystyle 12}$ ${\displaystyle -56}$ ${\displaystyle 72}$ ${\displaystyle 286}$ ${\displaystyle 42}$ ${\displaystyle -672}$ ${\displaystyle -{\frac {4688}{3}}}$ ${\displaystyle -{\frac {800}{3}}}$ ${\displaystyle -216}$ ${\displaystyle 288}$ ${\displaystyle 1568}$ ${\displaystyle 3432}$ ${\displaystyle 504}$ ${\displaystyle {\frac {87151}{10}}}$ ${\displaystyle {\frac {1694}{15}}}$ ${\displaystyle {\frac {52822}{15}}}$ ${\displaystyle {\frac {529}{6}}}$ ${\displaystyle {\frac {4431}{10}}}$

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 ${\displaystyle t^{r}q^{j}}$ are shown, along with their alternating sums ${\displaystyle \chi }$ (fixed ${\displaystyle j}$, alternation over ${\displaystyle r}$). The squares with yellow highlighting are those on the "critical diagonals", where ${\displaystyle j-2r=s+1}$ or ${\displaystyle j-2r=s-1}$, where ${\displaystyle s=}$-2 is the signature of K11n1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-9-8-7-6-5-4-3-2-10χ
-1         11
-3        110
-5       2  2
-7      21  -1
-9     22   0
-11    22    0
-13   22     0
-15  12      1
-17 12       -1
-19 1        1
-211         -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-3}$ ${\displaystyle i=-1}$ ${\displaystyle r=-9}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-8}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-7}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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