K11a124

K11a123

K11a125

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a124 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 3 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a124/ThurstonBennequinNumber Hyperbolic Volume 17.524 A-Polynomial See Data:K11a124/A-polynomial

[edit Notes for K11a124's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant -6

[edit Notes for K11a124's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$5t^{3}-18t^{2}+34t-41+34t^{-1}-18t^{-2}+5t^{-3}$
Conway polynomial	$5z^{6}+12z^{4}+7z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 155, 6 }
Jones polynomial	$-q^{14}+5q^{13}-11q^{12}+17q^{11}-23q^{10}+25q^{9}-25q^{8}+21q^{7}-14q^{6}+9q^{5}-3q^{4}+q^{3}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-6}+3z^{6}a^{-8}+z^{6}a^{-10}+3z^{4}a^{-6}+11z^{4}a^{-8}-z^{4}a^{-10}-z^{4}a^{-12}+3z^{2}a^{-6}+13z^{2}a^{-8}-9z^{2}a^{-10}+a^{-6}+5a^{-8}-7a^{-10}+2a^{-12}$
Kauffman polynomial (db, data sources)	$2z^{10}a^{-10}+2z^{10}a^{-12}+5z^{9}a^{-9}+13z^{9}a^{-11}+8z^{9}a^{-13}+6z^{8}a^{-8}+14z^{8}a^{-10}+21z^{8}a^{-12}+13z^{8}a^{-14}+3z^{7}a^{-7}-10z^{7}a^{-11}+4z^{7}a^{-13}+11z^{7}a^{-15}+z^{6}a^{-6}-14z^{6}a^{-8}-38z^{6}a^{-10}-45z^{6}a^{-12}-17z^{6}a^{-14}+5z^{6}a^{-16}-6z^{5}a^{-7}-20z^{5}a^{-9}-25z^{5}a^{-11}-27z^{5}a^{-13}-15z^{5}a^{-15}+z^{5}a^{-17}-3z^{4}a^{-6}+16z^{4}a^{-8}+34z^{4}a^{-10}+23z^{4}a^{-12}+4z^{4}a^{-14}-4z^{4}a^{-16}+3z^{3}a^{-7}+25z^{3}a^{-9}+34z^{3}a^{-11}+17z^{3}a^{-13}+5z^{3}a^{-15}+3z^{2}a^{-6}-13z^{2}a^{-8}-19z^{2}a^{-10}-3z^{2}a^{-12}-10za^{-9}-13za^{-11}-3za^{-13}-a^{-6}+5a^{-8}+7a^{-10}+2a^{-12}$
The A2 invariant	Data:K11a124/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a124/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a124 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["K11a124"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$5t^{3}-18t^{2}+34t-41+34t^{-1}-18t^{-2}+5t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$5z^{6}+12z^{4}+7z^{2}+1$

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]=

{ 155, 6 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{14}+5q^{13}-11q^{12}+17q^{11}-23q^{10}+25q^{9}-25q^{8}+21q^{7}-14q^{6}+9q^{5}-3q^{4}+q^{3}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{6}a^{-6}+3z^{6}a^{-8}+z^{6}a^{-10}+3z^{4}a^{-6}+11z^{4}a^{-8}-z^{4}a^{-10}-z^{4}a^{-12}+3z^{2}a^{-6}+13z^{2}a^{-8}-9z^{2}a^{-10}+a^{-6}+5a^{-8}-7a^{-10}+2a^{-12}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{10}a^{-10}+2z^{10}a^{-12}+5z^{9}a^{-9}+13z^{9}a^{-11}+8z^{9}a^{-13}+6z^{8}a^{-8}+14z^{8}a^{-10}+21z^{8}a^{-12}+13z^{8}a^{-14}+3z^{7}a^{-7}-10z^{7}a^{-11}+4z^{7}a^{-13}+11z^{7}a^{-15}+z^{6}a^{-6}-14z^{6}a^{-8}-38z^{6}a^{-10}-45z^{6}a^{-12}-17z^{6}a^{-14}+5z^{6}a^{-16}-6z^{5}a^{-7}-20z^{5}a^{-9}-25z^{5}a^{-11}-27z^{5}a^{-13}-15z^{5}a^{-15}+z^{5}a^{-17}-3z^{4}a^{-6}+16z^{4}a^{-8}+34z^{4}a^{-10}+23z^{4}a^{-12}+4z^{4}a^{-14}-4z^{4}a^{-16}+3z^{3}a^{-7}+25z^{3}a^{-9}+34z^{3}a^{-11}+17z^{3}a^{-13}+5z^{3}a^{-15}+3z^{2}a^{-6}-13z^{2}a^{-8}-19z^{2}a^{-10}-3z^{2}a^{-12}-10za^{-9}-13za^{-11}-3za^{-13}-a^{-6}+5a^{-8}+7a^{-10}+2a^{-12}$

"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["K11a124"];

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]=

{ $5t^{3}-18t^{2}+34t-41+34t^{-1}-18t^{-2}+5t^{-3}$, $-q^{14}+5q^{13}-11q^{12}+17q^{11}-23q^{10}+25q^{9}-25q^{8}+21q^{7}-14q^{6}+9q^{5}-3q^{4}+q^{3}$ }

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:

(7, 16)

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 $28$ $128$ $392$ ${\frac {2306}{3}}$ ${\frac {286}{3}}$ $3584$ ${\frac {15584}{3}}$ ${\frac {2528}{3}}$ $512$ ${\frac {10976}{3}}$ $8192$ ${\frac {64568}{3}}$ ${\frac {8008}{3}}$ ${\frac {1096537}{30}}$ ${\frac {40606}{15}}$ ${\frac {483314}{45}}$ ${\frac {3143}{18}}$ ${\frac {37177}{30}}$

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=$6 is the signature of K11a124. 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 10 11 χ 29                       1 -1 27                     4   4 25                   7 1   -6 23                 10 4     6 21               13 7       -6 19             12 10         2 17           13 13           0 15         8 12             -4 13       6 13               7 11     3 8                 -5 9     6                   6 7 1 3                     -2 5 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a123 K11a125