© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11a331 K11a333

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X14,3,15,4 X10,6,11,5 X18,7,19,8 X2,10,3,9 X22,11,1,12 X20,14,21,13 X4,15,5,16 X12,18,13,17 X8,19,9,20 X16,22,17,21

Gauss Code:

{1, -5, 2, -8, 3, -1, 4, -10, 5, -3, 6, -9, 7, -2, 8, -11, 9, -4, 10, -7, 11, -6}

DT (Dowker-Thistlethwaite) Code:

6 14 10 18 2 22 20 4 12 8 16

Alexander Polynomial:

t-4 - 7t-3 + 22t-2 - 40t-1 + 49 - 40t + 22t2 - 7t3 + t4

Conway Polynomial:

1 + z2 + z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{189, 0}

Jones Polynomial:

- q-5 + 5q-4 - 13q-3 + 21q-2 - 27q-1 + 32 - 30q + 26q2 - 19q3 + 10q4 - 4q5 + q6

Other knots (up to mirrors) with the same Jones Polynomial:

{...}

A2 (sl(3)) Invariant:

- q-14 + 3q-12 - 5q-10 + 2q-8 - 4q-4 + 9q-2 - 3 + 7q2 - 2q4 - 3q6 + 3q8 - 6q10 + 3q12 - q16 + q18

HOMFLY-PT Polynomial:

a-4 + 2a-4z2 + a-4z4 - 4a-2 - 7a-2z2 - 6a-2z4 - 2a-2z6 + 6 + 8z2 + 7z4 + 4z6 + z8 - 2a2 - 2a2z2 - 2a2z4 - a2z6

Kauffman Polynomial:

a-6z2 - 2a-6z4 + a-6z6 - 2a-5z + 6a-5z3 - 8a-5z5 + 4a-5z7 + a-4 - 4a-4z2 + 11a-4z4 - 15a-4z6 + 8a-4z8 - 6a-3z + 13a-3z3 - 6a-3z5 - 10a-3z7 + 9a-3z9 + 4a-2 - 16a-2z2 + 44a-2z4 - 53a-2z6 + 17a-2z8 + 4a-2z10 - 7a-1z + 15a-1z3 - 35a-1z7 + 23a-1z9 + 6 - 17z2 + 50z4 - 72z6 + 28z8 + 4z10 - 5az + 14az3 - 18az5 - 8az7 + 14az9 + 2a2 - 6a2z2 + 17a2z4 - 30a2z6 + 19a2z8 - 2a3z + 6a3z3 - 15a3z5 + 13a3z7 - 2a4z4 + 5a4z6 + a5z5

V2 and V3, the type 2 and 3 Vassiliev invariants:

{1, -1}

Khovanov Homology: (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 11332. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 13                       1 j = 11                     3   j = 9                   7 1   j = 7                 12 3     j = 5               14 7       j = 3             16 12         j = 1           16 14           j = -1         12 17             j = -3       9 15               j = -5     4 12                 j = -7   1 9                   j = -9   4                     j = -11 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Show[DrawMorseLink[Knot[11, Alternating, 332]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 332]]
Out[3]=	PD[X[6, 2, 7, 1], X[14, 3, 15, 4], X[10, 6, 11, 5], X[18, 7, 19, 8], > X[2, 10, 3, 9], X[22, 11, 1, 12], X[20, 14, 21, 13], X[4, 15, 5, 16], > X[12, 18, 13, 17], X[8, 19, 9, 20], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 332]]
Out[4]=	GaussCode[1, -5, 2, -8, 3, -1, 4, -10, 5, -3, 6, -9, 7, -2, 8, -11, 9, -4, 10, > -7, 11, -6]
In[5]:=	DTCode[Knot[11, Alternating, 332]]
Out[5]=	DTCode[6, 14, 10, 18, 2, 22, 20, 4, 12, 8, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 332]][t]
Out[6]=	-4 7 22 40 2 3 4 49 + t - -- + -- - -- - 40 t + 22 t - 7 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 332]][z]
Out[7]=	2 6 8 1 + z + z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 332]}
In[9]:=	{KnotDet[Knot[11, Alternating, 332]], KnotSignature[Knot[11, Alternating, 332]]}
Out[9]=	{189, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 332]][q]
Out[10]=	-5 5 13 21 27 2 3 4 5 6 32 - q + -- - -- + -- - -- - 30 q + 26 q - 19 q + 10 q - 4 q + q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 332]}
In[12]:=	A2Invariant[Knot[11, Alternating, 332]][q]
Out[12]=	-14 3 5 2 4 9 2 4 6 8 10 -3 - q + --- - --- + -- - -- + -- + 7 q - 2 q - 3 q + 3 q - 6 q + 12 10 8 4 2 q q q q q 12 16 18 > 3 q - q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 332]][a, z]
Out[13]=	2 2 4 4 -4 4 2 2 2 z 7 z 2 2 4 z 6 z 6 + a - -- - 2 a + 8 z + ---- - ---- - 2 a z + 7 z + -- - ---- - 2 4 2 4 2 a a a a a 6 2 4 6 2 z 2 6 8 > 2 a z + 4 z - ---- - a z + z 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 332]][a, z]
Out[14]=	2 2 -4 4 2 2 z 6 z 7 z 3 2 z 4 z 6 + a + -- + 2 a - --- - --- - --- - 5 a z - 2 a z - 17 z + -- - ---- - 2 5 3 a 6 4 a a a a a 2 3 3 3 4 16 z 2 2 6 z 13 z 15 z 3 3 3 4 2 z > ----- - 6 a z + ---- + ----- + ----- + 14 a z + 6 a z + 50 z - ---- + 2 5 3 a 6 a a a a 4 4 5 5 11 z 44 z 2 4 4 4 8 z 6 z 5 3 5 > ----- + ----- + 17 a z - 2 a z - ---- - ---- - 18 a z - 15 a z + 4 2 5 3 a a a a 6 6 6 7 7 5 5 6 z 15 z 53 z 2 6 4 6 4 z 10 z > a z - 72 z + -- - ----- - ----- - 30 a z + 5 a z + ---- - ----- - 6 4 2 5 3 a a a a a 7 8 8 9 35 z 7 3 7 8 8 z 17 z 2 8 9 z > ----- - 8 a z + 13 a z + 28 z + ---- + ----- + 19 a z + ---- + a 4 2 3 a a a 9 10 23 z 9 10 4 z > ----- + 14 a z + 4 z + ----- a 2 a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 332]], Vassiliev[3][Knot[11, Alternating, 332]]}
Out[15]=	{1, -1}
In[16]:=	Kh[Knot[11, Alternating, 332]][q, t]
Out[16]=	17 1 4 1 9 4 12 9 15 -- + 16 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t 12 3 3 2 5 2 5 3 7 3 > --- + 14 q t + 16 q t + 12 q t + 14 q t + 7 q t + 12 q t + q t 7 4 9 4 9 5 11 5 13 6 > 3 q t + 7 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a332

K11a331 K11a333