K11n111

K11n110

K11n112

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n111 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X5,14,6,15 X7,17,8,16 X9,19,10,18 X2,11,3,12 X13,21,14,20 X15,22,16,1 X17,9,18,8 X19,13,20,12 X21,7,22,6
Gauss code	1, -6, 2, -1, -3, 11, -4, 9, -5, -2, 6, 10, -7, 3, -8, 4, -9, 5, -10, 7, -11, 8
Dowker-Thistlethwaite code	4 10 -14 -16 -18 2 -20 -22 -8 -12 -6

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n111/ThurstonBennequinNumber Hyperbolic Volume 8.61249 A-Polynomial See Data:K11n111/A-polynomial

[edit Notes for K11n111'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 K11n111's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+t^{2}+3t-5+3t^{-1}+t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}-5z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 7, 2 }
Jones polynomial	$q^{7}-2q^{6}+2q^{5}-2q^{4}+2q^{3}-q^{2}+1-q^{-1}+q^{-2}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-2}-6z^{4}a^{-2}+z^{4}-9z^{2}a^{-2}+2z^{2}a^{-4}+z^{2}a^{-6}+4z^{2}-4a^{-2}+2a^{-4}+3$
Kauffman polynomial (db, data sources)	$z^{9}a^{-1}+z^{9}a^{-3}+2z^{8}a^{-2}+z^{8}a^{-4}+z^{8}-7z^{7}a^{-1}-7z^{7}a^{-3}-15z^{6}a^{-2}-7z^{6}a^{-4}+z^{6}a^{-6}-7z^{6}+14z^{5}a^{-1}+13z^{5}a^{-3}+z^{5}a^{-5}+2z^{5}a^{-7}+32z^{4}a^{-2}+14z^{4}a^{-4}-2z^{4}a^{-6}+z^{4}a^{-8}+15z^{4}-9z^{3}a^{-1}-6z^{3}a^{-3}-2z^{3}a^{-5}-5z^{3}a^{-7}-22z^{2}a^{-2}-8z^{2}a^{-4}-2z^{2}a^{-8}-12z^{2}+za^{-1}+za^{-3}+za^{-5}+za^{-7}+4a^{-2}+2a^{-4}+3$
The A2 invariant	$q^{6}+q^{4}+q^{2}+q^{-2}-q^{-4}-q^{-6}-q^{-10}+q^{-12}+q^{-16}-q^{-20}+q^{-22}$
The G2 invariant	Data:K11n111/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n111 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["K11n111"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+t^{2}+3t-5+3t^{-1}+t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{6}-5z^{4}-2z^{2}+1$

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]=

{ 7, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{7}-2q^{6}+2q^{5}-2q^{4}+2q^{3}-q^{2}+1-q^{-1}+q^{-2}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{6}a^{-2}-6z^{4}a^{-2}+z^{4}-9z^{2}a^{-2}+2z^{2}a^{-4}+z^{2}a^{-6}+4z^{2}-4a^{-2}+2a^{-4}+3$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-1}+z^{9}a^{-3}+2z^{8}a^{-2}+z^{8}a^{-4}+z^{8}-7z^{7}a^{-1}-7z^{7}a^{-3}-15z^{6}a^{-2}-7z^{6}a^{-4}+z^{6}a^{-6}-7z^{6}+14z^{5}a^{-1}+13z^{5}a^{-3}+z^{5}a^{-5}+2z^{5}a^{-7}+32z^{4}a^{-2}+14z^{4}a^{-4}-2z^{4}a^{-6}+z^{4}a^{-8}+15z^{4}-9z^{3}a^{-1}-6z^{3}a^{-3}-2z^{3}a^{-5}-5z^{3}a^{-7}-22z^{2}a^{-2}-8z^{2}a^{-4}-2z^{2}a^{-8}-12z^{2}+za^{-1}+za^{-3}+za^{-5}+za^{-7}+4a^{-2}+2a^{-4}+3$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

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["K11n111"];

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]=

{ $-t^{3}+t^{2}+3t-5+3t^{-1}+t^{-2}-t^{-3}$, $q^{7}-2q^{6}+2q^{5}-2q^{4}+2q^{3}-q^{2}+1-q^{-1}+q^{-2}$ }

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]=

{}

In[6]:=

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

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

Out[6]=

{}

Vassiliev invariants

V2 and V3:

(-2, -2)

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 $-8$ $-16$ $32$ ${\frac {212}{3}}$ ${\frac {172}{3}}$ $128$ ${\frac {992}{3}}$ ${\frac {320}{3}}$ $112$ $-{\frac {256}{3}}$ $128$ $-{\frac {1696}{3}}$ $-{\frac {1376}{3}}$ ${\frac {89}{15}}$ ${\frac {6044}{15}}$ $-{\frac {32284}{45}}$ ${\frac {2023}{9}}$ $-{\frac {3751}{15}}$

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 K11n111. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11n110 K11n112