K11a157

K11a156

K11a158

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a157 at Knotilus!

Knot presentations

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

A Braid Representative
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:K11a157/ThurstonBennequinNumber Hyperbolic Volume 16.8128 A-Polynomial See Data:K11a157/A-polynomial

[edit Notes for K11a157's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a157's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+6t^{3}-16t^{2}+28t-33+28t^{-1}-16t^{-2}+6t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-2z^{6}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\left\{2,t^{4}+t^{2}+1\right\}$
Determinant and Signature	{ 135, -2 }
Jones polynomial	$q^{3}-4q^{2}+9q-14+19q^{-1}-22q^{-2}+22q^{-3}-18q^{-4}+14q^{-5}-8q^{-6}+3q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-a^{2}z^{8}+2a^{4}z^{6}-5a^{2}z^{6}+z^{6}-a^{6}z^{4}+8a^{4}z^{4}-10a^{2}z^{4}+3z^{4}-3a^{6}z^{2}+12a^{4}z^{2}-10a^{2}z^{2}+3z^{2}-3a^{6}+7a^{4}-5a^{2}+2$
Kauffman polynomial (db, data sources)	$2a^{4}z^{10}+2a^{2}z^{10}+6a^{5}z^{9}+12a^{3}z^{9}+6az^{9}+8a^{6}z^{8}+13a^{4}z^{8}+12a^{2}z^{8}+7z^{8}+6a^{7}z^{7}-3a^{5}z^{7}-20a^{3}z^{7}-7az^{7}+4z^{7}a^{-1}+3a^{8}z^{6}-14a^{6}z^{6}-39a^{4}z^{6}-39a^{2}z^{6}+z^{6}a^{-2}-16z^{6}+a^{9}z^{5}-9a^{7}z^{5}-8a^{5}z^{5}+4a^{3}z^{5}-7az^{5}-9z^{5}a^{-1}-4a^{8}z^{4}+16a^{6}z^{4}+46a^{4}z^{4}+38a^{2}z^{4}-2z^{4}a^{-2}+10z^{4}-2a^{9}z^{3}+6a^{7}z^{3}+13a^{5}z^{3}+8a^{3}z^{3}+8az^{3}+5z^{3}a^{-1}+a^{8}z^{2}-12a^{6}z^{2}-28a^{4}z^{2}-20a^{2}z^{2}+z^{2}a^{-2}-4z^{2}+a^{9}z-2a^{7}z-5a^{5}z-3a^{3}z-2az-za^{-1}+3a^{6}+7a^{4}+5a^{2}+2$
The A2 invariant	$-q^{24}-q^{20}-3q^{18}+4q^{16}-q^{14}+4q^{12}+3q^{10}-3q^{8}+3q^{6}-6q^{4}+3q^{2}-q^{-2}+3q^{-4}-2q^{-6}+q^{-8}$
The G2 invariant	$q^{128}-2q^{126}+5q^{124}-8q^{122}+10q^{120}-10q^{118}+4q^{116}+10q^{114}-30q^{112}+53q^{110}-73q^{108}+74q^{106}-53q^{104}+89q^{100}-184q^{98}+263q^{96}-285q^{94}+205q^{92}-43q^{90}-200q^{88}+444q^{86}-594q^{84}+578q^{82}-353q^{80}-33q^{78}+442q^{76}-718q^{74}+741q^{72}-503q^{70}+62q^{68}+382q^{66}-634q^{64}+590q^{62}-242q^{60}-225q^{58}+619q^{56}-714q^{54}+457q^{52}+41q^{50}-598q^{48}+990q^{46}-1019q^{44}+687q^{42}-80q^{40}-565q^{38}+1029q^{36}-1139q^{34}+855q^{32}-321q^{30}-291q^{28}+734q^{26}-857q^{24}+643q^{22}-176q^{20}-321q^{18}+622q^{16}-611q^{14}+275q^{12}+201q^{10}-617q^{8}+783q^{6}-619q^{4}+209q^{2}+290-671q^{-2}+802q^{-4}-649q^{-6}+303q^{-8}+78q^{-10}-371q^{-12}+490q^{-14}-435q^{-16}+279q^{-18}-80q^{-20}-74q^{-22}+152q^{-24}-163q^{-26}+122q^{-28}-66q^{-30}+20q^{-32}+12q^{-34}-24q^{-36}+22q^{-38}-16q^{-40}+8q^{-42}-3q^{-44}+q^{-46}$

Further Quantum Invariants

K11a157 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["K11a157"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+6t^{3}-16t^{2}+28t-33+28t^{-1}-16t^{-2}+6t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

$\left\{2,t^{4}+t^{2}+1\right\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 135, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{3}-4q^{2}+9q-14+19q^{-1}-22q^{-2}+22q^{-3}-18q^{-4}+14q^{-5}-8q^{-6}+3q^{-7}-q^{-8}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2a^{4}z^{10}+2a^{2}z^{10}+6a^{5}z^{9}+12a^{3}z^{9}+6az^{9}+8a^{6}z^{8}+13a^{4}z^{8}+12a^{2}z^{8}+7z^{8}+6a^{7}z^{7}-3a^{5}z^{7}-20a^{3}z^{7}-7az^{7}+4z^{7}a^{-1}+3a^{8}z^{6}-14a^{6}z^{6}-39a^{4}z^{6}-39a^{2}z^{6}+z^{6}a^{-2}-16z^{6}+a^{9}z^{5}-9a^{7}z^{5}-8a^{5}z^{5}+4a^{3}z^{5}-7az^{5}-9z^{5}a^{-1}-4a^{8}z^{4}+16a^{6}z^{4}+46a^{4}z^{4}+38a^{2}z^{4}-2z^{4}a^{-2}+10z^{4}-2a^{9}z^{3}+6a^{7}z^{3}+13a^{5}z^{3}+8a^{3}z^{3}+8az^{3}+5z^{3}a^{-1}+a^{8}z^{2}-12a^{6}z^{2}-28a^{4}z^{2}-20a^{2}z^{2}+z^{2}a^{-2}-4z^{2}+a^{9}z-2a^{7}z-5a^{5}z-3a^{3}z-2az-za^{-1}+3a^{6}+7a^{4}+5a^{2}+2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a264, K11a305,}

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["K11a157"];

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]=

{ $-t^{4}+6t^{3}-16t^{2}+28t-33+28t^{-1}-16t^{-2}+6t^{-3}-t^{-4}$, $q^{3}-4q^{2}+9q-14+19q^{-1}-22q^{-2}+22q^{-3}-18q^{-4}+14q^{-5}-8q^{-6}+3q^{-7}-q^{-8}$ }

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]=

{K11a264, K11a305,}

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, -5)

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 $8$ $-40$ $32$ ${\frac {412}{3}}$ ${\frac {44}{3}}$ $-320$ $-{\frac {1840}{3}}$ $-{\frac {160}{3}}$ $-104$ ${\frac {256}{3}}$ $800$ ${\frac {3296}{3}}$ ${\frac {352}{3}}$ ${\frac {45871}{15}}$ $-{\frac {108}{5}}$ ${\frac {50644}{45}}$ ${\frac {929}{9}}$ ${\frac {1711}{15}}$

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 K11a157. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11a156 K11a158