K11n133

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 (Knotscape image) See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n133 at Knotilus!

Knot presentations

 Planar diagram presentation X4251 X10,4,11,3 X5,19,6,18 X7,20,8,21 X2,10,3,9 X11,17,12,16 X13,6,14,7 X15,9,16,8 X17,1,18,22 X19,15,20,14 X21,12,22,13 Gauss code 1, -5, 2, -1, -3, 7, -4, 8, 5, -2, -6, 11, -7, 10, -8, 6, -9, 3, -10, 4, -11, 9 Dowker-Thistlethwaite code 4 10 -18 -20 2 -16 -6 -8 -22 -14 -12

Three dimensional invariants

 Symmetry type Reversible Unknotting number ${\displaystyle \{2,3\}}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n133/ThurstonBennequinNumber Hyperbolic Volume 10.992 A-Polynomial See Data:K11n133/A-polynomial

Four dimensional invariants

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

Polynomial invariants

 Alexander polynomial ${\displaystyle t^{4}-4t^{3}+6t^{2}-2t-1-2t^{-1}+6t^{-2}-4t^{-3}+t^{-4}}$ Conway polynomial ${\displaystyle z^{8}+4z^{6}+2z^{4}+2z^{2}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{5,t+1\}}$ Determinant and Signature { 25, 4 } Jones polynomial ${\displaystyle -q^{7}+2q^{6}-3q^{5}+4q^{4}-4q^{3}+4q^{2}-3q+3-q^{-1}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle z^{8}a^{-4}-z^{6}a^{-2}+6z^{6}a^{-4}-z^{6}a^{-6}-4z^{4}a^{-2}+11z^{4}a^{-4}-5z^{4}a^{-6}-2z^{2}a^{-2}+8z^{2}a^{-4}-5z^{2}a^{-6}+z^{2}a^{-8}+a^{-2}+a^{-4}-a^{-6}}$ Kauffman polynomial (db, data sources) ${\displaystyle 2z^{9}a^{-3}+2z^{9}a^{-5}+3z^{8}a^{-2}+6z^{8}a^{-4}+3z^{8}a^{-6}+z^{7}a^{-1}-8z^{7}a^{-3}-8z^{7}a^{-5}+z^{7}a^{-7}-15z^{6}a^{-2}-31z^{6}a^{-4}-16z^{6}a^{-6}-4z^{5}a^{-1}+3z^{5}a^{-3}+2z^{5}a^{-5}-5z^{5}a^{-7}+19z^{4}a^{-2}+42z^{4}a^{-4}+23z^{4}a^{-6}+3z^{3}a^{-1}+7z^{3}a^{-3}+10z^{3}a^{-5}+6z^{3}a^{-7}-6z^{2}a^{-2}-16z^{2}a^{-4}-11z^{2}a^{-6}-z^{2}a^{-8}-za^{-1}-3za^{-3}-5za^{-5}-3za^{-7}-a^{-2}+a^{-4}+a^{-6}}$ The A2 invariant ${\displaystyle -q^{2}+1+q^{-2}+q^{-4}+q^{-6}+q^{-8}+q^{-10}-2q^{-12}+q^{-14}-q^{-16}+q^{-18}-2q^{-26}+q^{-28}}$ The G2 invariant Data:K11n133/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, ${\displaystyle q\leftrightarrow q^{-1}}$): {K11n50, K11n132,}

Vassiliev invariants

 V2 and V3: (2, 3)
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 ${\displaystyle 8}$ ${\displaystyle 24}$ ${\displaystyle 32}$ ${\displaystyle {\frac {220}{3}}}$ ${\displaystyle -{\frac {4}{3}}}$ ${\displaystyle 192}$ ${\displaystyle 304}$ ${\displaystyle 32}$ ${\displaystyle 24}$ ${\displaystyle {\frac {256}{3}}}$ ${\displaystyle 288}$ ${\displaystyle {\frac {1760}{3}}}$ ${\displaystyle -{\frac {32}{3}}}$ ${\displaystyle {\frac {18991}{15}}}$ ${\displaystyle {\frac {236}{15}}}$ ${\displaystyle {\frac {19204}{45}}}$ ${\displaystyle -{\frac {319}{9}}}$ ${\displaystyle {\frac {751}{15}}}$

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 ${\displaystyle t^{r}q^{j}}$ 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=}$4 is the signature of K11n133. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-3-2-1012345χ
15        1-1
13       211
11      21 -1
9     221 1
7    33   0
5   22    0
3  241    1
1 11      0
-1 2       2
-31        -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=1}$ ${\displaystyle i=3}$ ${\displaystyle i=5}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$

Computer Talk

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