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

K11a336 K11a338

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 14 16 18 20 22 4 2 12 10 8

Alexander Polynomial:

8t-2 - 22t-1 + 29 - 22t + 8t2

Conway Polynomial:

1 + 10z2 + 8z4

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{89, 4}

Jones Polynomial:

q2 - 3q3 + 7q4 - 10q5 + 13q6 - 14q7 + 14q8 - 11q9 + 8q10 - 5q11 + 2q12 - q13

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

{...}

A2 (sl(3)) Invariant:

q6 - 2q8 + 2q10 + q12 - 2q14 + 3q16 - q18 + q20 + 2q22 + 3q26 - 2q28 + q32 - 3q34 - q40

HOMFLY-PT Polynomial:

- a-12 - a-12z2 - a-10 + a-10z4 + 2a-8 + 5a-8z2 + 3a-8z4 + a-6 + 5a-6z2 + 3a-6z4 + a-4z2 + a-4z4

Kauffman Polynomial:

- 4a-15z + 8a-15z3 - 5a-15z5 + a-15z7 - 2a-14z2 + 9a-14z4 - 8a-14z6 + 2a-14z8 + 4a-13z3 - 5a-13z7 + 2a-13z9 - a-12 + 6a-12z2 - 7a-12z6 + a-12z8 + a-12z10 + 2a-11z - 5a-11z3 + 6a-11z5 - 11a-11z7 + 5a-11z9 + a-10 - 5a-10z2 + 7a-10z4 - 13a-10z6 + 5a-10z8 + a-10z10 - 2a-9z + 6a-9z3 - 11a-9z5 + 2a-9z7 + 3a-9z9 + 2a-8 - 6a-8z2 + 7a-8z4 - 8a-8z6 + 6a-8z8 + 5a-7z3 - 9a-7z5 + 7a-7z7 - a-6 + 6a-6z2 - 8a-6z4 + 6a-6z6 - 2a-5z3 + 3a-5z5 - a-4z2 + a-4z4

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

{10, 31}

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=4 is the signature of 11337. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 r = 10 r = 11 j = 27                       1 j = 25                     1   j = 23                   4 1   j = 21                 4 1     j = 19               7 4       j = 17             7 4         j = 15           7 7           j = 13         6 7             j = 11       4 7               j = 9     3 6                 j = 7     4                   j = 5 1 3                     j = 3 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, 337]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 337]]
Out[3]=	PD[X[6, 2, 7, 1], X[14, 4, 15, 3], X[16, 6, 17, 5], X[18, 8, 19, 7], > X[20, 10, 21, 9], X[22, 12, 1, 11], X[4, 14, 5, 13], X[2, 16, 3, 15], > X[12, 18, 13, 17], X[10, 20, 11, 19], X[8, 22, 9, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 337]]
Out[4]=	GaussCode[1, -8, 2, -7, 3, -1, 4, -11, 5, -10, 6, -9, 7, -2, 8, -3, 9, -4, 10, > -5, 11, -6]
In[5]:=	DTCode[Knot[11, Alternating, 337]]
Out[5]=	DTCode[6, 14, 16, 18, 20, 22, 4, 2, 12, 10, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 337]][t]
Out[6]=	8 22 2 29 + -- - -- - 22 t + 8 t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 337]][z]
Out[7]=	2 4 1 + 10 z + 8 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 337]}
In[9]:=	{KnotDet[Knot[11, Alternating, 337]], KnotSignature[Knot[11, Alternating, 337]]}
Out[9]=	{89, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 337]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 11 q - 3 q + 7 q - 10 q + 13 q - 14 q + 14 q - 11 q + 8 q - 5 q + 12 13 > 2 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 337]}
In[12]:=	A2Invariant[Knot[11, Alternating, 337]][q]
Out[12]=	6 8 10 12 14 16 18 20 22 26 28 q - 2 q + 2 q + q - 2 q + 3 q - q + q + 2 q + 3 q - 2 q + 32 34 40 > q - 3 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 337]][a, z]
Out[13]=	2 2 2 2 4 4 4 4 -12 -10 2 -6 z 5 z 5 z z z 3 z 3 z z -a - a + -- + a - --- + ---- + ---- + -- + --- + ---- + ---- + -- 8 12 8 6 4 10 8 6 4 a a a a a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 337]][a, z]
Out[14]=	2 2 2 2 2 -12 -10 2 -6 4 z 2 z 2 z 2 z 6 z 5 z 6 z 6 z -a + a + -- - a - --- + --- - --- - ---- + ---- - ---- - ---- + ---- - 8 15 11 9 14 12 10 8 6 a a a a a a a a a 2 3 3 3 3 3 3 4 4 4 4 z 8 z 4 z 5 z 6 z 5 z 2 z 9 z 7 z 7 z 8 z > -- + ---- + ---- - ---- + ---- + ---- - ---- + ---- + ---- + ---- - ---- + 4 15 13 11 9 7 5 14 10 8 6 a a a a a a a a a a a 4 5 5 5 5 5 6 6 6 6 z 5 z 6 z 11 z 9 z 3 z 8 z 7 z 13 z 8 z > -- - ---- + ---- - ----- - ---- + ---- - ---- - ---- - ----- - ---- + 4 15 11 9 7 5 14 12 10 8 a a a a a a a a a a 6 7 7 7 7 7 8 8 8 8 9 6 z z 5 z 11 z 2 z 7 z 2 z z 5 z 6 z 2 z > ---- + --- - ---- - ----- + ---- + ---- + ---- + --- + ---- + ---- + ---- + 6 15 13 11 9 7 14 12 10 8 13 a a a a a a a a a a a 9 9 10 10 5 z 3 z z z > ---- + ---- + --- + --- 11 9 12 10 a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 337]], Vassiliev[3][Knot[11, Alternating, 337]]}
Out[15]=	{10, 31}
In[16]:=	Kh[Knot[11, Alternating, 337]][q, t]
Out[16]=	3 5 5 7 2 9 2 9 3 11 3 11 4 q + q + 3 q t + 4 q t + 3 q t + 6 q t + 4 q t + 7 q t + 13 4 13 5 15 5 15 6 17 6 17 7 > 6 q t + 7 q t + 7 q t + 7 q t + 7 q t + 4 q t + 19 7 19 8 21 8 21 9 23 9 23 10 25 10 > 7 q t + 4 q t + 4 q t + q t + 4 q t + q t + q t + 27 11 > q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a337

K11a336 K11a338