K11a162

K11a161

K11a163

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a162 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X18,5,19,6 X14,7,15,8 X16,10,17,9 X2,11,3,12 X6,13,7,14 X20,16,21,15 X22,17,1,18 X12,20,13,19 X8,21,9,22
Gauss code	1, -6, 2, -1, 3, -7, 4, -11, 5, -2, 6, -10, 7, -4, 8, -5, 9, -3, 10, -8, 11, -9
Dowker-Thistlethwaite code	4 10 18 14 16 2 6 20 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:K11a162/ThurstonBennequinNumber Hyperbolic Volume 18.245 A-Polynomial See Data:K11a162/A-polynomial

[edit Notes for K11a162's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a162's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+7t^{3}-20t^{2}+35t-41+35t^{-1}-20t^{-2}+7t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-z^{6}+2z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 167, -2 }
Jones polynomial	$q^{3}-5q^{2}+11q-17+24q^{-1}-27q^{-2}+27q^{-3}-23q^{-4}+17q^{-5}-10q^{-6}+4q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-a^{2}z^{8}+2a^{4}z^{6}-4a^{2}z^{6}+z^{6}-a^{6}z^{4}+6a^{4}z^{4}-5a^{2}z^{4}+2z^{4}-2a^{6}z^{2}+5a^{4}z^{2}-a^{2}z^{2}-a^{6}+a^{4}+a^{2}$
Kauffman polynomial (db, data sources)	$3a^{4}z^{10}+3a^{2}z^{10}+9a^{5}z^{9}+18a^{3}z^{9}+9az^{9}+12a^{6}z^{8}+20a^{4}z^{8}+18a^{2}z^{8}+10z^{8}+9a^{7}z^{7}-2a^{5}z^{7}-27a^{3}z^{7}-11az^{7}+5z^{7}a^{-1}+4a^{8}z^{6}-18a^{6}z^{6}-53a^{4}z^{6}-54a^{2}z^{6}+z^{6}a^{-2}-22z^{6}+a^{9}z^{5}-13a^{7}z^{5}-18a^{5}z^{5}-2a^{3}z^{5}-7az^{5}-9z^{5}a^{-1}-4a^{8}z^{4}+13a^{6}z^{4}+42a^{4}z^{4}+39a^{2}z^{4}-z^{4}a^{-2}+13z^{4}-a^{9}z^{3}+9a^{7}z^{3}+18a^{5}z^{3}+13a^{3}z^{3}+8az^{3}+3z^{3}a^{-1}+a^{8}z^{2}-6a^{6}z^{2}-13a^{4}z^{2}-7a^{2}z^{2}-z^{2}-3a^{7}z-5a^{5}z-3a^{3}z-az+a^{6}+a^{4}-a^{2}$
The A2 invariant	$-q^{24}+q^{22}-3q^{18}+4q^{16}-4q^{14}+2q^{12}+2q^{10}-3q^{8}+6q^{6}-5q^{4}+5q^{2}-2q^{-2}+3q^{-4}-3q^{-6}+q^{-8}$
The G2 invariant	$q^{128}-3q^{126}+7q^{124}-13q^{122}+17q^{120}-19q^{118}+12q^{116}+11q^{114}-46q^{112}+94q^{110}-138q^{108}+151q^{106}-118q^{104}+17q^{102}+152q^{100}-344q^{98}+509q^{96}-563q^{94}+427q^{92}-103q^{90}-372q^{88}+858q^{86}-1159q^{84}+1122q^{82}-679q^{80}-70q^{78}+863q^{76}-1399q^{74}+1437q^{72}-936q^{70}+69q^{68}+791q^{66}-1265q^{64}+1134q^{62}-429q^{60}-505q^{58}+1243q^{56}-1419q^{54}+900q^{52}+100q^{50}-1198q^{48}+1938q^{46}-1983q^{44}+1306q^{42}-122q^{40}-1126q^{38}+1989q^{36}-2169q^{34}+1607q^{32}-550q^{30}-614q^{28}+1451q^{26}-1645q^{24}+1181q^{22}-254q^{20}-695q^{18}+1248q^{16}-1177q^{14}+500q^{12}+447q^{10}-1235q^{8}+1534q^{6}-1190q^{4}+367q^{2}+595-1310q^{-2}+1523q^{-4}-1208q^{-6}+535q^{-8}+190q^{-10}-721q^{-12}+919q^{-14}-794q^{-16}+480q^{-18}-114q^{-20}-162q^{-22}+286q^{-24}-290q^{-26}+206q^{-28}-102q^{-30}+23q^{-32}+28q^{-34}-42q^{-36}+36q^{-38}-24q^{-40}+11q^{-42}-4q^{-44}+q^{-46}$

Further Quantum Invariants

K11a162 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["K11a162"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+7t^{3}-20t^{2}+35t-41+35t^{-1}-20t^{-2}+7t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 167, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{3}-5q^{2}+11q-17+24q^{-1}-27q^{-2}+27q^{-3}-23q^{-4}+17q^{-5}-10q^{-6}+4q^{-7}-q^{-8}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-a^{2}z^{8}+2a^{4}z^{6}-4a^{2}z^{6}+z^{6}-a^{6}z^{4}+6a^{4}z^{4}-5a^{2}z^{4}+2z^{4}-2a^{6}z^{2}+5a^{4}z^{2}-a^{2}z^{2}-a^{6}+a^{4}+a^{2}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$3a^{4}z^{10}+3a^{2}z^{10}+9a^{5}z^{9}+18a^{3}z^{9}+9az^{9}+12a^{6}z^{8}+20a^{4}z^{8}+18a^{2}z^{8}+10z^{8}+9a^{7}z^{7}-2a^{5}z^{7}-27a^{3}z^{7}-11az^{7}+5z^{7}a^{-1}+4a^{8}z^{6}-18a^{6}z^{6}-53a^{4}z^{6}-54a^{2}z^{6}+z^{6}a^{-2}-22z^{6}+a^{9}z^{5}-13a^{7}z^{5}-18a^{5}z^{5}-2a^{3}z^{5}-7az^{5}-9z^{5}a^{-1}-4a^{8}z^{4}+13a^{6}z^{4}+42a^{4}z^{4}+39a^{2}z^{4}-z^{4}a^{-2}+13z^{4}-a^{9}z^{3}+9a^{7}z^{3}+18a^{5}z^{3}+13a^{3}z^{3}+8az^{3}+3z^{3}a^{-1}+a^{8}z^{2}-6a^{6}z^{2}-13a^{4}z^{2}-7a^{2}z^{2}-z^{2}-3a^{7}z-5a^{5}z-3a^{3}z-az+a^{6}+a^{4}-a^{2}$

"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["K11a162"];

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]=

{ $-t^{4}+7t^{3}-20t^{2}+35t-41+35t^{-1}-20t^{-2}+7t^{-3}-t^{-4}$, $q^{3}-5q^{2}+11q-17+24q^{-1}-27q^{-2}+27q^{-3}-23q^{-4}+17q^{-5}-10q^{-6}+4q^{-7}-q^{-8}$ }

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]=

{}

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, -3)

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 $8$ $-24$ $32$ ${\frac {220}{3}}$ $-{\frac {4}{3}}$ $-192$ $-304$ $0$ $-56$ ${\frac {256}{3}}$ $288$ ${\frac {1760}{3}}$ $-{\frac {32}{3}}$ ${\frac {18991}{15}}$ $-{\frac {724}{15}}$ ${\frac {16324}{45}}$ ${\frac {833}{9}}$ ${\frac {271}{15}}$

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 K11a162. 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                     4   -4 3                   7 1   6 1                 10 4     -6 -1               14 7       7 -3             14 11         -3 -5           13 13           0 -7         10 14             4 -9       7 13               -6 -11     3 10                 7 -13   1 7                   -6 -15   3                     3 -17 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-4$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-3$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-2$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=-1$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{14}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{14}$ $r=1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a161 K11a163