K11a146

K11a145

K11a147

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a146 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 1 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a146/ThurstonBennequinNumber Hyperbolic Volume 16.2882 A-Polynomial See Data:K11a146/A-polynomial

[edit Notes for K11a146'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 K11a146's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+6t^{3}-15t^{2}+25t-29+25t^{-1}-15t^{-2}+6t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-2z^{6}+z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 123, -2 }
Jones polynomial	$q^{3}-4q^{2}+8q-13+18q^{-1}-19q^{-2}+20q^{-3}-17q^{-4}+12q^{-5}-7q^{-6}+3q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-a^{2}z^{8}+2a^{4}z^{6}-5a^{2}z^{6}+z^{6}-a^{6}z^{4}+8a^{4}z^{4}-9a^{2}z^{4}+3z^{4}-3a^{6}z^{2}+10a^{4}z^{2}-6a^{2}z^{2}+2z^{2}-2a^{6}+3a^{4}$
Kauffman polynomial (db, data sources)	$2a^{4}z^{10}+2a^{2}z^{10}+5a^{5}z^{9}+11a^{3}z^{9}+6az^{9}+6a^{6}z^{8}+8a^{4}z^{8}+9a^{2}z^{8}+7z^{8}+5a^{7}z^{7}-4a^{5}z^{7}-24a^{3}z^{7}-11az^{7}+4z^{7}a^{-1}+3a^{8}z^{6}-8a^{6}z^{6}-26a^{4}z^{6}-35a^{2}z^{6}+z^{6}a^{-2}-19z^{6}+a^{9}z^{5}-7a^{7}z^{5}-a^{5}z^{5}+18a^{3}z^{5}+az^{5}-10z^{5}a^{-1}-5a^{8}z^{4}+6a^{6}z^{4}+31a^{4}z^{4}+37a^{2}z^{4}-2z^{4}a^{-2}+15z^{4}-2a^{9}z^{3}+3a^{7}z^{3}+4a^{5}z^{3}-3a^{3}z^{3}+3az^{3}+5z^{3}a^{-1}+2a^{8}z^{2}-5a^{6}z^{2}-16a^{4}z^{2}-14a^{2}z^{2}-5z^{2}+a^{9}z-a^{7}z-2a^{5}z+2a^{6}+3a^{4}$
The A2 invariant	$-q^{24}-2q^{18}+3q^{16}-3q^{14}+q^{12}+2q^{10}-q^{8}+6q^{6}-3q^{4}+3q^{2}-1-2q^{-2}+2q^{-4}-2q^{-6}+q^{-8}$
The G2 invariant	$q^{128}-2q^{126}+5q^{124}-8q^{122}+9q^{120}-8q^{118}+q^{116}+13q^{114}-29q^{112}+47q^{110}-58q^{108}+51q^{106}-27q^{104}-22q^{102}+82q^{100}-138q^{98}+174q^{96}-174q^{94}+120q^{92}-12q^{90}-135q^{88}+288q^{86}-385q^{84}+378q^{82}-253q^{80}+10q^{78}+262q^{76}-476q^{74}+541q^{72}-397q^{70}+105q^{68}+222q^{66}-443q^{64}+447q^{62}-240q^{60}-93q^{58}+389q^{56}-504q^{54}+369q^{52}-22q^{50}-379q^{48}+680q^{46}-732q^{44}+507q^{42}-92q^{40}-371q^{38}+716q^{36}-819q^{34}+660q^{32}-282q^{30}-157q^{28}+513q^{26}-649q^{24}+524q^{22}-206q^{20}-166q^{18}+428q^{16}-470q^{14}+276q^{12}+73q^{10}-400q^{8}+570q^{6}-493q^{4}+193q^{2}+180-489q^{-2}+609q^{-4}-510q^{-6}+255q^{-8}+49q^{-10}-289q^{-12}+393q^{-14}-355q^{-16}+223q^{-18}-65q^{-20}-62q^{-22}+125q^{-24}-135q^{-26}+103q^{-28}-54q^{-30}+17q^{-32}+11q^{-34}-20q^{-36}+18q^{-38}-14q^{-40}+7q^{-42}-3q^{-44}+q^{-46}$

Further Quantum Invariants

K11a146 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["K11a146"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+6t^{3}-15t^{2}+25t-29+25t^{-1}-15t^{-2}+6t^{-3}-t^{-4}$

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]=

{ 123, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{3}-4q^{2}+8q-13+18q^{-1}-19q^{-2}+20q^{-3}-17q^{-4}+12q^{-5}-7q^{-6}+3q^{-7}-q^{-8}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2a^{4}z^{10}+2a^{2}z^{10}+5a^{5}z^{9}+11a^{3}z^{9}+6az^{9}+6a^{6}z^{8}+8a^{4}z^{8}+9a^{2}z^{8}+7z^{8}+5a^{7}z^{7}-4a^{5}z^{7}-24a^{3}z^{7}-11az^{7}+4z^{7}a^{-1}+3a^{8}z^{6}-8a^{6}z^{6}-26a^{4}z^{6}-35a^{2}z^{6}+z^{6}a^{-2}-19z^{6}+a^{9}z^{5}-7a^{7}z^{5}-a^{5}z^{5}+18a^{3}z^{5}+az^{5}-10z^{5}a^{-1}-5a^{8}z^{4}+6a^{6}z^{4}+31a^{4}z^{4}+37a^{2}z^{4}-2z^{4}a^{-2}+15z^{4}-2a^{9}z^{3}+3a^{7}z^{3}+4a^{5}z^{3}-3a^{3}z^{3}+3az^{3}+5z^{3}a^{-1}+2a^{8}z^{2}-5a^{6}z^{2}-16a^{4}z^{2}-14a^{2}z^{2}-5z^{2}+a^{9}z-a^{7}z-2a^{5}z+2a^{6}+3a^{4}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a294,}

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["K11a146"];

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]=

{ $-t^{4}+6t^{3}-15t^{2}+25t-29+25t^{-1}-15t^{-2}+6t^{-3}-t^{-4}$, $q^{3}-4q^{2}+8q-13+18q^{-1}-19q^{-2}+20q^{-3}-17q^{-4}+12q^{-5}-7q^{-6}+3q^{-7}-q^{-8}$ }

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]=

{K11a294,}

Vassiliev invariants

V2 and V3:

(3, -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 $12$ $-40$ $72$ $174$ $26$ $-480$ $-{\frac {2512}{3}}$ $-{\frac {256}{3}}$ $-168$ $288$ $800$ $2088$ $312$ ${\frac {41151}{10}}$ $-{\frac {1826}{15}}$ ${\frac {8594}{5}}$ ${\frac {235}{2}}$ ${\frac {2271}{10}}$

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 $s=$-2 is the signature of K11a146. 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             10 9         -1 -5           10 9           1 -7         7 10             3 -9       5 10               -5 -11     2 7                 5 -13   1 5                   -4 -15   2                     2 -17 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

K11a145 K11a147