K11a123

K11a122

K11a124

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a123 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{2,3\}$ 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a123/ThurstonBennequinNumber Hyperbolic Volume 15.9713 A-Polynomial See Data:K11a123/A-polynomial

[edit Notes for K11a123's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a123's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$9t^{2}-29t+41-29t^{-1}+9t^{-2}$
Conway polynomial	$9z^{4}+7z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{3,t+1\}$
Determinant and Signature	{ 117, 4 }
Jones polynomial	$-q^{13}+4q^{12}-8q^{11}+12q^{10}-17q^{9}+18q^{8}-18q^{7}+17q^{6}-11q^{5}+7q^{4}-3q^{3}+q^{2}$
HOMFLY-PT polynomial (db, data sources)	$z^{4}a^{-4}+3z^{4}a^{-6}+4z^{4}a^{-8}+z^{4}a^{-10}+z^{2}a^{-4}+4z^{2}a^{-6}+6z^{2}a^{-8}-3z^{2}a^{-10}-z^{2}a^{-12}+2a^{-6}+2a^{-8}-4a^{-10}+a^{-12}$
Kauffman polynomial (db, data sources)	$2z^{10}a^{-10}+2z^{10}a^{-12}+6z^{9}a^{-9}+11z^{9}a^{-11}+5z^{9}a^{-13}+9z^{8}a^{-8}+11z^{8}a^{-10}+6z^{8}a^{-12}+4z^{8}a^{-14}+8z^{7}a^{-7}-2z^{7}a^{-9}-25z^{7}a^{-11}-14z^{7}a^{-13}+z^{7}a^{-15}+6z^{6}a^{-6}-14z^{6}a^{-8}-40z^{6}a^{-10}-34z^{6}a^{-12}-14z^{6}a^{-14}+3z^{5}a^{-5}-9z^{5}a^{-7}-20z^{5}a^{-9}+z^{5}a^{-11}+6z^{5}a^{-13}-3z^{5}a^{-15}+z^{4}a^{-4}-7z^{4}a^{-6}+8z^{4}a^{-8}+33z^{4}a^{-10}+31z^{4}a^{-12}+14z^{4}a^{-14}-2z^{3}a^{-5}+4z^{3}a^{-7}+23z^{3}a^{-9}+20z^{3}a^{-11}+6z^{3}a^{-13}+3z^{3}a^{-15}-z^{2}a^{-4}+6z^{2}a^{-6}-2z^{2}a^{-8}-12z^{2}a^{-10}-6z^{2}a^{-12}-3z^{2}a^{-14}-9za^{-9}-12za^{-11}-4za^{-13}-za^{-15}-2a^{-6}+2a^{-8}+4a^{-10}+a^{-12}$
The A2 invariant	Data:K11a123/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a123/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a123 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["K11a123"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$9t^{2}-29t+41-29t^{-1}+9t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$9z^{4}+7z^{2}+1$

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]=

$\{3,t+1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 117, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{13}+4q^{12}-8q^{11}+12q^{10}-17q^{9}+18q^{8}-18q^{7}+17q^{6}-11q^{5}+7q^{4}-3q^{3}+q^{2}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{10}a^{-10}+2z^{10}a^{-12}+6z^{9}a^{-9}+11z^{9}a^{-11}+5z^{9}a^{-13}+9z^{8}a^{-8}+11z^{8}a^{-10}+6z^{8}a^{-12}+4z^{8}a^{-14}+8z^{7}a^{-7}-2z^{7}a^{-9}-25z^{7}a^{-11}-14z^{7}a^{-13}+z^{7}a^{-15}+6z^{6}a^{-6}-14z^{6}a^{-8}-40z^{6}a^{-10}-34z^{6}a^{-12}-14z^{6}a^{-14}+3z^{5}a^{-5}-9z^{5}a^{-7}-20z^{5}a^{-9}+z^{5}a^{-11}+6z^{5}a^{-13}-3z^{5}a^{-15}+z^{4}a^{-4}-7z^{4}a^{-6}+8z^{4}a^{-8}+33z^{4}a^{-10}+31z^{4}a^{-12}+14z^{4}a^{-14}-2z^{3}a^{-5}+4z^{3}a^{-7}+23z^{3}a^{-9}+20z^{3}a^{-11}+6z^{3}a^{-13}+3z^{3}a^{-15}-z^{2}a^{-4}+6z^{2}a^{-6}-2z^{2}a^{-8}-12z^{2}a^{-10}-6z^{2}a^{-12}-3z^{2}a^{-14}-9za^{-9}-12za^{-11}-4za^{-13}-za^{-15}-2a^{-6}+2a^{-8}+4a^{-10}+a^{-12}$

"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["K11a123"];

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]=

{ $9t^{2}-29t+41-29t^{-1}+9t^{-2}$, $-q^{13}+4q^{12}-8q^{11}+12q^{10}-17q^{9}+18q^{8}-18q^{7}+17q^{6}-11q^{5}+7q^{4}-3q^{3}+q^{2}$ }

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:

(7, 17)

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 $28$ $136$ $392$ ${\frac {2594}{3}}$ ${\frac {358}{3}}$ $3808$ ${\frac {18352}{3}}$ ${\frac {3040}{3}}$ $712$ ${\frac {10976}{3}}$ $9248$ ${\frac {72632}{3}}$ ${\frac {10024}{3}}$ ${\frac {1339417}{30}}$ ${\frac {33766}{15}}$ ${\frac {678794}{45}}$ ${\frac {5447}{18}}$ ${\frac {55897}{30}}$

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=$4 is the signature of K11a123. 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 27                       1 -1 25                     3   3 23                   5 1   -4 21                 7 3     4 19               10 5       -5 17             8 7         1 15           10 10           0 13         7 8             -1 11       4 10               6 9     3 7                 -4 7     4                   4 5 1 3                     -2 3 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a122 K11a124