K11a151

K11a150

K11a152

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a151 at Knotilus!

Knot presentations

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

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:K11a151/ThurstonBennequinNumber Hyperbolic Volume 16.1582 A-Polynomial See Data:K11a151/A-polynomial

[edit Notes for K11a151'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 K11a151's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+6t^{3}-15t^{2}+26t-31+26t^{-1}-15t^{-2}+6t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-2z^{6}+z^{4}+4z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 127, -2 }
Jones polynomial	$q^{3}-4q^{2}+8q-13+18q^{-1}-20q^{-2}+21q^{-3}-17q^{-4}+13q^{-5}-8q^{-6}+3q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-a^{2}z^{8}+2a^{4}z^{6}-5a^{2}z^{6}+z^{6}-a^{6}z^{4}+8a^{4}z^{4}-9a^{2}z^{4}+3z^{4}-3a^{6}z^{2}+11a^{4}z^{2}-6a^{2}z^{2}+2z^{2}-3a^{6}+5a^{4}-a^{2}$
Kauffman polynomial (db, data sources)	$2a^{4}z^{10}+2a^{2}z^{10}+5a^{5}z^{9}+11a^{3}z^{9}+6az^{9}+7a^{6}z^{8}+9a^{4}z^{8}+9a^{2}z^{8}+7z^{8}+6a^{7}z^{7}-21a^{3}z^{7}-11az^{7}+4z^{7}a^{-1}+3a^{8}z^{6}-10a^{6}z^{6}-25a^{4}z^{6}-32a^{2}z^{6}+z^{6}a^{-2}-19z^{6}+a^{9}z^{5}-10a^{7}z^{5}-13a^{5}z^{5}+10a^{3}z^{5}+2az^{5}-10z^{5}a^{-1}-4a^{8}z^{4}+8a^{6}z^{4}+25a^{4}z^{4}+30a^{2}z^{4}-2z^{4}a^{-2}+15z^{4}-2a^{9}z^{3}+8a^{7}z^{3}+17a^{5}z^{3}+5a^{3}z^{3}+3az^{3}+5z^{3}a^{-1}+a^{8}z^{2}-7a^{6}z^{2}-14a^{4}z^{2}-10a^{2}z^{2}-4z^{2}+a^{9}z-4a^{7}z-8a^{5}z-4a^{3}z-az+3a^{6}+5a^{4}+a^{2}$
The A2 invariant	$-q^{24}-q^{20}-3q^{18}+3q^{16}-2q^{14}+3q^{12}+3q^{10}-q^{8}+5q^{6}-4q^{4}+3q^{2}-1-2q^{-2}+2q^{-4}-2q^{-6}+q^{-8}$
The G2 invariant	$q^{128}-2q^{126}+5q^{124}-8q^{122}+10q^{120}-10q^{118}+4q^{116}+11q^{114}-32q^{112}+56q^{110}-75q^{108}+70q^{106}-41q^{104}-19q^{102}+105q^{100}-184q^{98}+239q^{96}-232q^{94}+140q^{92}+12q^{90}-210q^{88}+385q^{86}-475q^{84}+426q^{82}-233q^{80}-67q^{78}+367q^{76}-556q^{74}+551q^{72}-349q^{70}+12q^{68}+307q^{66}-480q^{64}+418q^{62}-138q^{60}-217q^{58}+498q^{56}-544q^{54}+320q^{52}+92q^{50}-523q^{48}+802q^{46}-791q^{44}+499q^{42}-7q^{40}-488q^{38}+825q^{36}-877q^{34}+640q^{32}-209q^{30}-256q^{28}+578q^{26}-650q^{24}+469q^{22}-106q^{20}-264q^{18}+479q^{16}-454q^{14}+191q^{12}+181q^{10}-496q^{8}+612q^{6}-476q^{4}+143q^{2}+247-544q^{-2}+639q^{-4}-511q^{-6}+234q^{-8}+73q^{-10}-308q^{-12}+400q^{-14}-354q^{-16}+222q^{-18}-62q^{-20}-65q^{-22}+127q^{-24}-135q^{-26}+102q^{-28}-54q^{-30}+17q^{-32}+11q^{-34}-20q^{-36}+18q^{-38}-14q^{-40}+7q^{-42}-3q^{-44}+q^{-46}$

Further Quantum Invariants

K11a151 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["K11a151"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+6t^{3}-15t^{2}+26t-31+26t^{-1}-15t^{-2}+6t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 127, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{3}-4q^{2}+8q-13+18q^{-1}-20q^{-2}+21q^{-3}-17q^{-4}+13q^{-5}-8q^{-6}+3q^{-7}-q^{-8}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-a^{2}z^{8}+2a^{4}z^{6}-5a^{2}z^{6}+z^{6}-a^{6}z^{4}+8a^{4}z^{4}-9a^{2}z^{4}+3z^{4}-3a^{6}z^{2}+11a^{4}z^{2}-6a^{2}z^{2}+2z^{2}-3a^{6}+5a^{4}-a^{2}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2a^{4}z^{10}+2a^{2}z^{10}+5a^{5}z^{9}+11a^{3}z^{9}+6az^{9}+7a^{6}z^{8}+9a^{4}z^{8}+9a^{2}z^{8}+7z^{8}+6a^{7}z^{7}-21a^{3}z^{7}-11az^{7}+4z^{7}a^{-1}+3a^{8}z^{6}-10a^{6}z^{6}-25a^{4}z^{6}-32a^{2}z^{6}+z^{6}a^{-2}-19z^{6}+a^{9}z^{5}-10a^{7}z^{5}-13a^{5}z^{5}+10a^{3}z^{5}+2az^{5}-10z^{5}a^{-1}-4a^{8}z^{4}+8a^{6}z^{4}+25a^{4}z^{4}+30a^{2}z^{4}-2z^{4}a^{-2}+15z^{4}-2a^{9}z^{3}+8a^{7}z^{3}+17a^{5}z^{3}+5a^{3}z^{3}+3az^{3}+5z^{3}a^{-1}+a^{8}z^{2}-7a^{6}z^{2}-14a^{4}z^{2}-10a^{2}z^{2}-4z^{2}+a^{9}z-4a^{7}z-8a^{5}z-4a^{3}z-az+3a^{6}+5a^{4}+a^{2}$

"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["K11a151"];

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]=

{ $-t^{4}+6t^{3}-15t^{2}+26t-31+26t^{-1}-15t^{-2}+6t^{-3}-t^{-4}$, $q^{3}-4q^{2}+8q-13+18q^{-1}-20q^{-2}+21q^{-3}-17q^{-4}+13q^{-5}-8q^{-6}+3q^{-7}-q^{-8}$ }

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]=

{}

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, -7)

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 $16$ $-56$ $128$ ${\frac {872}{3}}$ ${\frac {160}{3}}$ $-896$ $-{\frac {4784}{3}}$ $-{\frac {608}{3}}$ $-344$ ${\frac {2048}{3}}$ $1568$ ${\frac {13952}{3}}$ ${\frac {2560}{3}}$ ${\frac {132422}{15}}$ $-{\frac {8848}{15}}$ ${\frac {196688}{45}}$ ${\frac {1786}{9}}$ ${\frac {9782}{15}}$

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 K11a151. 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                   5 1   4 1                 8 3     -5 -1               10 5       5 -3             11 9         -2 -5           10 9           1 -7         7 11             4 -9       6 10               -4 -11     2 7                 5 -13   1 6                   -5 -15   2                     2 -17 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

K11a150 K11a152