K11a111

K11a110

K11a112

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a111 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,4,11,3 X14,5,15,6 X16,8,17,7 X2,10,3,9 X22,11,1,12 X20,13,21,14 X18,16,19,15 X8,18,9,17 X6,19,7,20 X12,21,13,22
Gauss code	1, -5, 2, -1, 3, -10, 4, -9, 5, -2, 6, -11, 7, -3, 8, -4, 9, -8, 10, -7, 11, -6
Dowker-Thistlethwaite code	4 10 14 16 2 22 20 18 8 6 12

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a111/ThurstonBennequinNumber Hyperbolic Volume 13.6745 A-Polynomial See Data:K11a111/A-polynomial

[edit Notes for K11a111's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant -2

[edit Notes for K11a111's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-10t^{2}+24t-31+24t^{-1}-10t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}+2z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 103, 2 }
Jones polynomial	$q^{7}-4q^{6}+7q^{5}-11q^{4}+15q^{3}-16q^{2}+16q-13+10q^{-1}-6q^{-2}+3q^{-3}-q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+z^{6}-a^{2}z^{4}+2z^{4}a^{-2}-2z^{4}a^{-4}+3z^{4}-2a^{2}z^{2}+2z^{2}a^{-2}-3z^{2}a^{-4}+z^{2}a^{-6}+4z^{2}-a^{2}+a^{-2}-a^{-4}+2$
Kauffman polynomial (db, data sources)	$z^{10}a^{-2}+z^{10}+3az^{9}+7z^{9}a^{-1}+4z^{9}a^{-3}+3a^{2}z^{8}+11z^{8}a^{-2}+7z^{8}a^{-4}+7z^{8}+a^{3}z^{7}-7az^{7}-13z^{7}a^{-1}+3z^{7}a^{-3}+8z^{7}a^{-5}-12a^{2}z^{6}-33z^{6}a^{-2}-5z^{6}a^{-4}+7z^{6}a^{-6}-33z^{6}-4a^{3}z^{5}-3az^{5}-7z^{5}a^{-1}-19z^{5}a^{-3}-7z^{5}a^{-5}+4z^{5}a^{-7}+15a^{2}z^{4}+24z^{4}a^{-2}-6z^{4}a^{-4}-7z^{4}a^{-6}+z^{4}a^{-8}+37z^{4}+5a^{3}z^{3}+12az^{3}+17z^{3}a^{-1}+12z^{3}a^{-3}-z^{3}a^{-5}-3z^{3}a^{-7}-7a^{2}z^{2}-4z^{2}a^{-2}+5z^{2}a^{-4}+2z^{2}a^{-6}-14z^{2}-2a^{3}z-5az-5za^{-1}-za^{-3}+za^{-5}+a^{2}-a^{-2}-a^{-4}+2$
The A2 invariant	Data:K11a111/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a111/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a111 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["K11a111"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-10t^{2}+24t-31+24t^{-1}-10t^{-2}+2t^{-3}$

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]=

{ 103, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{7}-4q^{6}+7q^{5}-11q^{4}+15q^{3}-16q^{2}+16q-13+10q^{-1}-6q^{-2}+3q^{-3}-q^{-4}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{6}a^{-2}+z^{6}-a^{2}z^{4}+2z^{4}a^{-2}-2z^{4}a^{-4}+3z^{4}-2a^{2}z^{2}+2z^{2}a^{-2}-3z^{2}a^{-4}+z^{2}a^{-6}+4z^{2}-a^{2}+a^{-2}-a^{-4}+2$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-2}+z^{10}+3az^{9}+7z^{9}a^{-1}+4z^{9}a^{-3}+3a^{2}z^{8}+11z^{8}a^{-2}+7z^{8}a^{-4}+7z^{8}+a^{3}z^{7}-7az^{7}-13z^{7}a^{-1}+3z^{7}a^{-3}+8z^{7}a^{-5}-12a^{2}z^{6}-33z^{6}a^{-2}-5z^{6}a^{-4}+7z^{6}a^{-6}-33z^{6}-4a^{3}z^{5}-3az^{5}-7z^{5}a^{-1}-19z^{5}a^{-3}-7z^{5}a^{-5}+4z^{5}a^{-7}+15a^{2}z^{4}+24z^{4}a^{-2}-6z^{4}a^{-4}-7z^{4}a^{-6}+z^{4}a^{-8}+37z^{4}+5a^{3}z^{3}+12az^{3}+17z^{3}a^{-1}+12z^{3}a^{-3}-z^{3}a^{-5}-3z^{3}a^{-7}-7a^{2}z^{2}-4z^{2}a^{-2}+5z^{2}a^{-4}+2z^{2}a^{-6}-14z^{2}-2a^{3}z-5az-5za^{-1}-za^{-3}+za^{-5}+a^{2}-a^{-2}-a^{-4}+2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_117, K11a23,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a68,}

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["K11a111"];

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]=

{ $2t^{3}-10t^{2}+24t-31+24t^{-1}-10t^{-2}+2t^{-3}$, $q^{7}-4q^{6}+7q^{5}-11q^{4}+15q^{3}-16q^{2}+16q-13+10q^{-1}-6q^{-2}+3q^{-3}-q^{-4}$ }

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_117, K11a23,}

In[6]:=

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

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

Out[6]=

{K11a68,}

Vassiliev invariants

V2 and V3:

(2, 1)

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 $8$ $8$ $32$ ${\frac {124}{3}}$ $-{\frac {4}{3}}$ $64$ ${\frac {368}{3}}$ ${\frac {128}{3}}$ $8$ ${\frac {256}{3}}$ $32$ ${\frac {992}{3}}$ $-{\frac {32}{3}}$ ${\frac {6751}{15}}$ ${\frac {796}{15}}$ ${\frac {4804}{45}}$ $-{\frac {31}{9}}$ ${\frac {271}{15}}$

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=$2 is the signature of K11a111. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11a110 K11a112