K11n43

K11n42

K11n44

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n43 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X5,13,6,12 X2837 X18,10,19,9 X20,11,21,12 X13,7,14,6 X10,16,11,15 X22,18,1,17 X14,19,15,20 X16,22,17,21
Gauss code	1, -4, 2, -1, -3, 7, 4, -2, 5, -8, 6, 3, -7, -10, 8, -11, 9, -5, 10, -6, 11, -9
Dowker-Thistlethwaite code	4 8 -12 2 18 20 -6 10 22 14 16

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n43/ThurstonBennequinNumber Hyperbolic Volume 15.7945 A-Polynomial See Data:K11n43/A-polynomial

[edit Notes for K11n43's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant -4

[edit Notes for K11n43's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+10t^{2}-20t+25-20t^{-1}+10t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}-2z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 89, 4 }
Jones polynomial	$-q^{11}+4q^{10}-8q^{9}+12q^{8}-15q^{7}+15q^{6}-14q^{5}+11q^{4}-6q^{3}+3q^{2}$
HOMFLY-PT polynomial (db, data sources)	$-2z^{6}a^{-6}+3z^{4}a^{-4}-8z^{4}a^{-6}+3z^{4}a^{-8}+8z^{2}a^{-4}-12z^{2}a^{-6}+7z^{2}a^{-8}-z^{2}a^{-10}+5a^{-4}-7a^{-6}+4a^{-8}-a^{-10}$
Kauffman polynomial (db, data sources)	$2z^{9}a^{-7}+2z^{9}a^{-9}+5z^{8}a^{-6}+11z^{8}a^{-8}+6z^{8}a^{-10}+3z^{7}a^{-5}+7z^{7}a^{-7}+11z^{7}a^{-9}+7z^{7}a^{-11}-11z^{6}a^{-6}-20z^{6}a^{-8}-5z^{6}a^{-10}+4z^{6}a^{-12}-3z^{5}a^{-5}-20z^{5}a^{-7}-30z^{5}a^{-9}-12z^{5}a^{-11}+z^{5}a^{-13}+6z^{4}a^{-4}+21z^{4}a^{-6}+16z^{4}a^{-8}-5z^{4}a^{-10}-6z^{4}a^{-12}+4z^{3}a^{-5}+20z^{3}a^{-7}+23z^{3}a^{-9}+6z^{3}a^{-11}-z^{3}a^{-13}-11z^{2}a^{-4}-21z^{2}a^{-6}-10z^{2}a^{-8}+2z^{2}a^{-10}+2z^{2}a^{-12}-4za^{-5}-8za^{-7}-6za^{-9}-2za^{-11}+5a^{-4}+7a^{-6}+4a^{-8}+a^{-10}$
The A2 invariant	$3q^{-6}-q^{-8}+4q^{-10}+q^{-12}-2q^{-14}+2q^{-16}-5q^{-18}+q^{-20}-2q^{-22}+3q^{-26}-2q^{-28}+2q^{-30}-q^{-34}$
The G2 invariant	Data:K11n43/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n43 Quantum Invariants.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11n43"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$-2t^{3}+10t^{2}-20t+25-20t^{-1}+10t^{-2}-2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$-2z^{6}-2z^{4}+2z^{2}+1$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 89, 4 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$-q^{11}+4q^{10}-8q^{9}+12q^{8}-15q^{7}+15q^{6}-14q^{5}+11q^{4}-6q^{3}+3q^{2}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$-2z^{6}a^{-6}+3z^{4}a^{-4}-8z^{4}a^{-6}+3z^{4}a^{-8}+8z^{2}a^{-4}-12z^{2}a^{-6}+7z^{2}a^{-8}-z^{2}a^{-10}+5a^{-4}-7a^{-6}+4a^{-8}-a^{-10}$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$2z^{9}a^{-7}+2z^{9}a^{-9}+5z^{8}a^{-6}+11z^{8}a^{-8}+6z^{8}a^{-10}+3z^{7}a^{-5}+7z^{7}a^{-7}+11z^{7}a^{-9}+7z^{7}a^{-11}-11z^{6}a^{-6}-20z^{6}a^{-8}-5z^{6}a^{-10}+4z^{6}a^{-12}-3z^{5}a^{-5}-20z^{5}a^{-7}-30z^{5}a^{-9}-12z^{5}a^{-11}+z^{5}a^{-13}+6z^{4}a^{-4}+21z^{4}a^{-6}+16z^{4}a^{-8}-5z^{4}a^{-10}-6z^{4}a^{-12}+4z^{3}a^{-5}+20z^{3}a^{-7}+23z^{3}a^{-9}+6z^{3}a^{-11}-z^{3}a^{-13}-11z^{2}a^{-4}-21z^{2}a^{-6}-10z^{2}a^{-8}+2z^{2}a^{-10}+2z^{2}a^{-12}-4za^{-5}-8za^{-7}-6za^{-9}-2za^{-11}+5a^{-4}+7a^{-6}+4a^{-8}+a^{-10}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_92, K11a153, K11a224, K11n35,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n35,}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11n43"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]=

{ $-2t^{3}+10t^{2}-20t+25-20t^{-1}+10t^{-2}-2t^{-3}$, $-q^{11}+4q^{10}-8q^{9}+12q^{8}-15q^{7}+15q^{6}-14q^{5}+11q^{4}-6q^{3}+3q^{2}$ }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{10_92, K11a153, K11a224, K11n35,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{K11n35,}

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 $8$ $24$ $32$ ${\frac {364}{3}}$ ${\frac {92}{3}}$ $192$ $592$ $128$ $120$ ${\frac {256}{3}}$ $288$ ${\frac {2912}{3}}$ ${\frac {736}{3}}$ ${\frac {39991}{15}}$ $-{\frac {268}{5}}$ ${\frac {62644}{45}}$ ${\frac {377}{9}}$ ${\frac {2551}{15}}$

V_2,1 through V_6,9 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 $t^{r}q^{j}$ 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=$4 is the signature of K11n43. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ 0 1 2 3 4 5 6 7 8 9 χ 23                   1 -1 21                 3   3 19               5 1   -4 17             7 3     4 15           8 5       -3 13         7 7         0 11       7 8           1 9     4 7             -3 7   2 7               5 5 1 4                 -3 3 3                   3

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=3$ $i=5$ $r=0$ ${\mathbb {Z} }^{3}$ ${\mathbb {Z} }$ $r=1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=4$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=5$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=6$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=7$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=8$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=9$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

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

Modifying This Page

Read me first: Modifying Knot Pages. See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate). See/edit the Hoste-Thistlethwaite_Splice_Base (expert). Back to the top.

K11n42 K11n44