K11a114

K11a113

K11a115

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a114 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a114/ThurstonBennequinNumber Hyperbolic Volume 17.5584 A-Polynomial See Data:K11a114/A-polynomial

[edit Notes for K11a114'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 K11a114's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$3t^{3}-15t^{2}+35t-45+35t^{-1}-15t^{-2}+3t^{-3}$
Conway polynomial	$3z^{6}+3z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 151, 2 }
Jones polynomial	$q^{9}-4q^{8}+9q^{7}-16q^{6}+21q^{5}-24q^{4}+25q^{3}-21q^{2}+16q-9+4q^{-1}-q^{-2}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+2z^{6}a^{-4}+z^{4}a^{-2}+6z^{4}a^{-4}-3z^{4}a^{-6}-z^{4}+8z^{2}a^{-4}-6z^{2}a^{-6}+z^{2}a^{-8}-z^{2}+4a^{-4}-4a^{-6}+a^{-8}$
Kauffman polynomial (db, data sources)	$2z^{10}a^{-4}+2z^{10}a^{-6}+7z^{9}a^{-3}+13z^{9}a^{-5}+6z^{9}a^{-7}+10z^{8}a^{-2}+19z^{8}a^{-4}+16z^{8}a^{-6}+7z^{8}a^{-8}+8z^{7}a^{-1}+2z^{7}a^{-3}-12z^{7}a^{-5}-2z^{7}a^{-7}+4z^{7}a^{-9}-13z^{6}a^{-2}-46z^{6}a^{-4}-44z^{6}a^{-6}-14z^{6}a^{-8}+z^{6}a^{-10}+4z^{6}+az^{5}-11z^{5}a^{-1}-20z^{5}a^{-3}-18z^{5}a^{-5}-19z^{5}a^{-7}-9z^{5}a^{-9}+7z^{4}a^{-2}+37z^{4}a^{-4}+35z^{4}a^{-6}+8z^{4}a^{-8}-2z^{4}a^{-10}-5z^{4}-az^{3}+6z^{3}a^{-1}+16z^{3}a^{-3}+22z^{3}a^{-5}+20z^{3}a^{-7}+7z^{3}a^{-9}-2z^{2}a^{-2}-16z^{2}a^{-4}-15z^{2}a^{-6}-2z^{2}a^{-8}+z^{2}a^{-10}+2z^{2}-za^{-1}-3za^{-3}-7za^{-5}-7za^{-7}-2za^{-9}+4a^{-4}+4a^{-6}+a^{-8}$
The A2 invariant	Data:K11a114/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a114/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a114 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["K11a114"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$3t^{3}-15t^{2}+35t-45+35t^{-1}-15t^{-2}+3t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$3z^{6}+3z^{4}+2z^{2}+1$

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]=

{ 151, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{9}-4q^{8}+9q^{7}-16q^{6}+21q^{5}-24q^{4}+25q^{3}-21q^{2}+16q-9+4q^{-1}-q^{-2}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{6}a^{-2}+2z^{6}a^{-4}+z^{4}a^{-2}+6z^{4}a^{-4}-3z^{4}a^{-6}-z^{4}+8z^{2}a^{-4}-6z^{2}a^{-6}+z^{2}a^{-8}-z^{2}+4a^{-4}-4a^{-6}+a^{-8}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{10}a^{-4}+2z^{10}a^{-6}+7z^{9}a^{-3}+13z^{9}a^{-5}+6z^{9}a^{-7}+10z^{8}a^{-2}+19z^{8}a^{-4}+16z^{8}a^{-6}+7z^{8}a^{-8}+8z^{7}a^{-1}+2z^{7}a^{-3}-12z^{7}a^{-5}-2z^{7}a^{-7}+4z^{7}a^{-9}-13z^{6}a^{-2}-46z^{6}a^{-4}-44z^{6}a^{-6}-14z^{6}a^{-8}+z^{6}a^{-10}+4z^{6}+az^{5}-11z^{5}a^{-1}-20z^{5}a^{-3}-18z^{5}a^{-5}-19z^{5}a^{-7}-9z^{5}a^{-9}+7z^{4}a^{-2}+37z^{4}a^{-4}+35z^{4}a^{-6}+8z^{4}a^{-8}-2z^{4}a^{-10}-5z^{4}-az^{3}+6z^{3}a^{-1}+16z^{3}a^{-3}+22z^{3}a^{-5}+20z^{3}a^{-7}+7z^{3}a^{-9}-2z^{2}a^{-2}-16z^{2}a^{-4}-15z^{2}a^{-6}-2z^{2}a^{-8}+z^{2}a^{-10}+2z^{2}-za^{-1}-3za^{-3}-7za^{-5}-7za^{-7}-2za^{-9}+4a^{-4}+4a^{-6}+a^{-8}$

"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["K11a114"];

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]=

{ $3t^{3}-15t^{2}+35t-45+35t^{-1}-15t^{-2}+3t^{-3}$, $q^{9}-4q^{8}+9q^{7}-16q^{6}+21q^{5}-24q^{4}+25q^{3}-21q^{2}+16q-9+4q^{-1}-q^{-2}$ }

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 $8$ $16$ $32$ ${\frac {76}{3}}$ $-{\frac {28}{3}}$ $128$ ${\frac {160}{3}}$ ${\frac {64}{3}}$ $-48$ ${\frac {256}{3}}$ $128$ ${\frac {608}{3}}$ $-{\frac {224}{3}}$ ${\frac {1111}{15}}$ ${\frac {3316}{15}}$ $-{\frac {13076}{45}}$ $-{\frac {439}{9}}$ $-{\frac {809}{15}}$

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 K11a114. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 19                       1 1 17                     3   -3 15                   6 1   5 13                 10 3     -7 11               11 6       5 9             13 10         -3 7           12 11           1 5         9 13             4 3       7 12               -5 1     3 10                 7 -1   1 6                   -5 -3   3                     3 -5 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{7}$ $r=1$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=3$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=4$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=5$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a113 K11a115