K11a126

K11a125

K11a127

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a126 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2,3\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a126/ThurstonBennequinNumber Hyperbolic Volume 17.2092 A-Polynomial See Data:K11a126/A-polynomial

[edit Notes for K11a126's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $4$ Rasmussen s-Invariant 0

[edit Notes for K11a126's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-5t^{3}+16t^{2}-31t+39-31t^{-1}+16t^{-2}-5t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+3z^{6}+6z^{4}+4z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 145, 0 }
Jones polynomial	$q^{6}-5q^{5}+10q^{4}-16q^{3}+21q^{2}-23q+24-19q^{-1}+14q^{-2}-8q^{-3}+3q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}-a^{2}z^{6}-2z^{6}a^{-2}+6z^{6}-4a^{2}z^{4}-7z^{4}a^{-2}+z^{4}a^{-4}+16z^{4}-7a^{2}z^{2}-9z^{2}a^{-2}+z^{2}a^{-4}+19z^{2}-4a^{2}-4a^{-2}+9$
Kauffman polynomial (db, data sources)	$2z^{10}a^{-2}+2z^{10}+6az^{9}+13z^{9}a^{-1}+7z^{9}a^{-3}+8a^{2}z^{8}+17z^{8}a^{-2}+9z^{8}a^{-4}+16z^{8}+6a^{3}z^{7}-16z^{7}a^{-1}-5z^{7}a^{-3}+5z^{7}a^{-5}+3a^{4}z^{6}-13a^{2}z^{6}-50z^{6}a^{-2}-20z^{6}a^{-4}+z^{6}a^{-6}-45z^{6}+a^{5}z^{5}-9a^{3}z^{5}-16az^{5}-13z^{5}a^{-1}-17z^{5}a^{-3}-10z^{5}a^{-5}-4a^{4}z^{4}+15a^{2}z^{4}+41z^{4}a^{-2}+11z^{4}a^{-4}-z^{4}a^{-6}+48z^{4}-2a^{5}z^{3}+7a^{3}z^{3}+23az^{3}+25z^{3}a^{-1}+15z^{3}a^{-3}+4z^{3}a^{-5}+a^{4}z^{2}-11a^{2}z^{2}-17z^{2}a^{-2}-2z^{2}a^{-4}-27z^{2}+a^{5}z-3a^{3}z-10az-10za^{-1}-3za^{-3}+za^{-5}+4a^{2}+4a^{-2}+9$
The A2 invariant	$-q^{14}+q^{12}-4q^{10}+q^{8}+q^{6}-3q^{4}+7q^{2}-1+5q^{-2}+q^{-4}-2q^{-6}+3q^{-8}-5q^{-10}+q^{-12}-2q^{-16}+q^{-18}$
The G2 invariant	$q^{80}-2q^{78}+5q^{76}-8q^{74}+10q^{72}-10q^{70}+4q^{68}+10q^{66}-29q^{64}+52q^{62}-73q^{60}+75q^{58}-57q^{56}+5q^{54}+81q^{52}-176q^{50}+264q^{48}-301q^{46}+243q^{44}-90q^{42}-162q^{40}+435q^{38}-642q^{36}+679q^{34}-495q^{32}+97q^{30}+391q^{28}-790q^{26}+940q^{24}-756q^{22}+286q^{20}+280q^{18}-717q^{16}+822q^{14}-537q^{12}+9q^{10}+548q^{8}-837q^{6}+717q^{4}-214q^{2}-464+1042q^{-2}-1252q^{-4}+998q^{-6}-336q^{-8}-472q^{-10}+1152q^{-12}-1438q^{-14}+1242q^{-16}-639q^{-18}-146q^{-20}+810q^{-22}-1127q^{-24}+1002q^{-26}-494q^{-28}-155q^{-30}+664q^{-32}-825q^{-34}+565q^{-36}-24q^{-38}-560q^{-40}+909q^{-42}-872q^{-44}+456q^{-46}+157q^{-48}-722q^{-50}+1015q^{-52}-936q^{-54}+549q^{-56}-32q^{-58}-426q^{-60}+665q^{-62}-654q^{-64}+456q^{-66}-172q^{-68}-78q^{-70}+221q^{-72}-254q^{-74}+198q^{-76}-107q^{-78}+32q^{-80}+21q^{-82}-39q^{-84}+35q^{-86}-24q^{-88}+11q^{-90}-4q^{-92}+q^{-94}$

Further Quantum Invariants

K11a126 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["K11a126"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-5t^{3}+16t^{2}-31t+39-31t^{-1}+16t^{-2}-5t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{8}+3z^{6}+6z^{4}+4z^{2}+1$

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]=

{ 145, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{6}-5q^{5}+10q^{4}-16q^{3}+21q^{2}-23q+24-19q^{-1}+14q^{-2}-8q^{-3}+3q^{-4}-q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}-a^{2}z^{6}-2z^{6}a^{-2}+6z^{6}-4a^{2}z^{4}-7z^{4}a^{-2}+z^{4}a^{-4}+16z^{4}-7a^{2}z^{2}-9z^{2}a^{-2}+z^{2}a^{-4}+19z^{2}-4a^{2}-4a^{-2}+9$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{10}a^{-2}+2z^{10}+6az^{9}+13z^{9}a^{-1}+7z^{9}a^{-3}+8a^{2}z^{8}+17z^{8}a^{-2}+9z^{8}a^{-4}+16z^{8}+6a^{3}z^{7}-16z^{7}a^{-1}-5z^{7}a^{-3}+5z^{7}a^{-5}+3a^{4}z^{6}-13a^{2}z^{6}-50z^{6}a^{-2}-20z^{6}a^{-4}+z^{6}a^{-6}-45z^{6}+a^{5}z^{5}-9a^{3}z^{5}-16az^{5}-13z^{5}a^{-1}-17z^{5}a^{-3}-10z^{5}a^{-5}-4a^{4}z^{4}+15a^{2}z^{4}+41z^{4}a^{-2}+11z^{4}a^{-4}-z^{4}a^{-6}+48z^{4}-2a^{5}z^{3}+7a^{3}z^{3}+23az^{3}+25z^{3}a^{-1}+15z^{3}a^{-3}+4z^{3}a^{-5}+a^{4}z^{2}-11a^{2}z^{2}-17z^{2}a^{-2}-2z^{2}a^{-4}-27z^{2}+a^{5}z-3a^{3}z-10az-10za^{-1}-3za^{-3}+za^{-5}+4a^{2}+4a^{-2}+9$

"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["K11a126"];

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]=

{ $t^{4}-5t^{3}+16t^{2}-31t+39-31t^{-1}+16t^{-2}-5t^{-3}+t^{-4}$, $q^{6}-5q^{5}+10q^{4}-16q^{3}+21q^{2}-23q+24-19q^{-1}+14q^{-2}-8q^{-3}+3q^{-4}-q^{-5}$ }

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:

(4, 0)

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 $16$ $0$ $128$ ${\frac {440}{3}}$ ${\frac {40}{3}}$ $0$ $0$ $32$ $-32$ ${\frac {2048}{3}}$ $0$ ${\frac {7040}{3}}$ ${\frac {640}{3}}$ ${\frac {34622}{15}}$ ${\frac {1144}{5}}$ ${\frac {26648}{45}}$ ${\frac {322}{9}}$ ${\frac {542}{15}}$

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=$0 is the signature of K11a126. 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 13                       1 1 11                     4   -4 9                   6 1   5 7                 10 4     -6 5               11 6       5 3             12 10         -2 1           12 11           1 -1         8 13             5 -3       6 11               -5 -5     2 8                 6 -7   1 6                   -5 -9   2                     2 -11 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

K11a125 K11a127