K11a112

K11a111

K11a113

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a112 at Knotilus!

Knot presentations

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a112/ThurstonBennequinNumber Hyperbolic Volume 16.343 A-Polynomial See Data:K11a112/A-polynomial

[edit Notes for K11a112's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $1$ Rasmussen s-Invariant 0

[edit Notes for K11a112's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-6t^{3}+15t^{2}-25t+31-25t^{-1}+15t^{-2}-6t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+2z^{6}-z^{4}-3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 125, 0 }
Jones polynomial	$-q^{5}+4q^{4}-8q^{3}+14q^{2}-18q+20-20q^{-1}+17q^{-2}-12q^{-3}+7q^{-4}-3q^{-5}+q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}-2a^{2}z^{6}-z^{6}a^{-2}+5z^{6}+a^{4}z^{4}-8a^{2}z^{4}-3z^{4}a^{-2}+9z^{4}+3a^{4}z^{2}-10a^{2}z^{2}-2z^{2}a^{-2}+6z^{2}+2a^{4}-3a^{2}+a^{-2}+1$
Kauffman polynomial (db, data sources)	$2a^{2}z^{10}+2z^{10}+5a^{3}z^{9}+11az^{9}+6z^{9}a^{-1}+5a^{4}z^{8}+7a^{2}z^{8}+8z^{8}a^{-2}+10z^{8}+3a^{5}z^{7}-10a^{3}z^{7}-24az^{7}-4z^{7}a^{-1}+7z^{7}a^{-3}+a^{6}z^{6}-13a^{4}z^{6}-30a^{2}z^{6}-10z^{6}a^{-2}+4z^{6}a^{-4}-30z^{6}-8a^{5}z^{5}+6a^{3}z^{5}+19az^{5}-6z^{5}a^{-1}-10z^{5}a^{-3}+z^{5}a^{-5}-3a^{6}z^{4}+12a^{4}z^{4}+39a^{2}z^{4}-6z^{4}a^{-4}+30z^{4}+5a^{5}z^{3}-2a^{3}z^{3}-3az^{3}+8z^{3}a^{-1}+3z^{3}a^{-3}-z^{3}a^{-5}+2a^{6}z^{2}-8a^{4}z^{2}-21a^{2}z^{2}+3z^{2}a^{-2}+2z^{2}a^{-4}-10z^{2}-a^{5}z-az-3za^{-1}-za^{-3}+2a^{4}+3a^{2}-a^{-2}+1$
The A2 invariant	$q^{18}+2q^{12}-3q^{10}+3q^{8}-q^{6}-2q^{4}+2q^{2}-5+4q^{-2}-2q^{-4}+2q^{-6}+3q^{-8}-2q^{-10}+2q^{-12}-q^{-14}$
The G2 invariant	$q^{94}-2q^{92}+5q^{90}-9q^{88}+11q^{86}-12q^{84}+5q^{82}+11q^{80}-32q^{78}+59q^{76}-80q^{74}+82q^{72}-56q^{70}-11q^{68}+115q^{66}-222q^{64}+297q^{62}-283q^{60}+156q^{58}+70q^{56}-331q^{54}+533q^{52}-570q^{50}+403q^{48}-73q^{46}-309q^{44}+581q^{42}-627q^{40}+434q^{38}-68q^{36}-299q^{34}+502q^{32}-461q^{30}+179q^{28}+200q^{26}-506q^{24}+596q^{22}-418q^{20}+36q^{18}+410q^{16}-746q^{14}+838q^{12}-646q^{10}+216q^{8}+289q^{6}-705q^{4}+874q^{2}-737+363q^{-2}+107q^{-4}-487q^{-6}+626q^{-8}-488q^{-10}+146q^{-12}+231q^{-14}-463q^{-16}+455q^{-18}-206q^{-20}-151q^{-22}+463q^{-24}-584q^{-26}+485q^{-28}-208q^{-30}-138q^{-32}+410q^{-34}-527q^{-36}+476q^{-38}-288q^{-40}+60q^{-42}+137q^{-44}-258q^{-46}+281q^{-48}-233q^{-50}+141q^{-52}-39q^{-54}-40q^{-56}+83q^{-58}-94q^{-60}+78q^{-62}-47q^{-64}+20q^{-66}+3q^{-68}-14q^{-70}+15q^{-72}-13q^{-74}+7q^{-76}-3q^{-78}+q^{-80}$

Further Quantum Invariants

K11a112 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["K11a112"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-6t^{3}+15t^{2}-25t+31-25t^{-1}+15t^{-2}-6t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 125, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{5}+4q^{4}-8q^{3}+14q^{2}-18q+20-20q^{-1}+17q^{-2}-12q^{-3}+7q^{-4}-3q^{-5}+q^{-6}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}-2a^{2}z^{6}-z^{6}a^{-2}+5z^{6}+a^{4}z^{4}-8a^{2}z^{4}-3z^{4}a^{-2}+9z^{4}+3a^{4}z^{2}-10a^{2}z^{2}-2z^{2}a^{-2}+6z^{2}+2a^{4}-3a^{2}+a^{-2}+1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2a^{2}z^{10}+2z^{10}+5a^{3}z^{9}+11az^{9}+6z^{9}a^{-1}+5a^{4}z^{8}+7a^{2}z^{8}+8z^{8}a^{-2}+10z^{8}+3a^{5}z^{7}-10a^{3}z^{7}-24az^{7}-4z^{7}a^{-1}+7z^{7}a^{-3}+a^{6}z^{6}-13a^{4}z^{6}-30a^{2}z^{6}-10z^{6}a^{-2}+4z^{6}a^{-4}-30z^{6}-8a^{5}z^{5}+6a^{3}z^{5}+19az^{5}-6z^{5}a^{-1}-10z^{5}a^{-3}+z^{5}a^{-5}-3a^{6}z^{4}+12a^{4}z^{4}+39a^{2}z^{4}-6z^{4}a^{-4}+30z^{4}+5a^{5}z^{3}-2a^{3}z^{3}-3az^{3}+8z^{3}a^{-1}+3z^{3}a^{-3}-z^{3}a^{-5}+2a^{6}z^{2}-8a^{4}z^{2}-21a^{2}z^{2}+3z^{2}a^{-2}+2z^{2}a^{-4}-10z^{2}-a^{5}z-az-3za^{-1}-za^{-3}+2a^{4}+3a^{2}-a^{-2}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a5,}

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["K11a112"];

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]=

{ $t^{4}-6t^{3}+15t^{2}-25t+31-25t^{-1}+15t^{-2}-6t^{-3}+t^{-4}$, $-q^{5}+4q^{4}-8q^{3}+14q^{2}-18q+20-20q^{-1}+17q^{-2}-12q^{-3}+7q^{-4}-3q^{-5}+q^{-6}$ }

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]=

{K11a5,}

Vassiliev invariants

V2 and V3:

(-3, 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 $-12$ $16$ $72$ $98$ $46$ $-192$ $-{\frac {1088}{3}}$ $-{\frac {320}{3}}$ $-80$ $-288$ $128$ $-1176$ $-552$ $-{\frac {6991}{10}}$ ${\frac {862}{5}}$ $-{\frac {3994}{5}}$ ${\frac {293}{2}}$ $-{\frac {2031}{10}}$

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 K11a112. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     3   3 7                   5 1   -4 5                 9 3     6 3               9 5       -4 1             11 9         2 -1           10 10           0 -3         7 10             -3 -5       5 10               5 -7     2 7                 -5 -9   1 5                   4 -11   2                     -2 -13 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a111 K11a113