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

K11a333 K11a335

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 X2,14,3,13 X4,16,5,15 X12,18,13,17 X8,20,9,19 X10,22,11,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 14 16 18 20 22 2 4 12 8 10

Alexander Polynomial:

2t-4 - 4t-3 + 6t-2 - 8t-1 + 9 - 8t + 6t2 - 4t3 + 2t4

Conway Polynomial:

1 + 12z2 + 22z4 + 12z6 + 2z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{49, 8}

Jones Polynomial:

q4 - q5 + 3q6 - 4q7 + 6q8 - 7q9 + 7q10 - 7q11 + 6q12 - 4q13 + 2q14 - q15

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

{...}

A2 (sl(3)) Invariant:

q14 + 2q18 + 2q22 + q24 + q28 - 2q30 + q32 - q34 - q40 - q44

HOMFLY-PT Polynomial:

- 2a-12 - 7a-12z2 - 5a-12z4 - a-12z6 + 6a-10z2 + 11a-10z4 + 6a-10z6 + a-10z8 + 3a-8 + 13a-8z2 + 16a-8z4 + 7a-8z6 + a-8z8

Kauffman Polynomial:

- a-19z + a-19z3 - a-18z2 + 2a-18z4 + a-17z - 2a-17z3 + 3a-17z5 + 4a-16z2 - 6a-16z4 + 4a-16z6 - a-15z + 5a-15z3 - 8a-15z5 + 4a-15z7 + a-14z4 - 6a-14z6 + 3a-14z8 + a-13z - 4a-13z3 + 3a-13z5 - 5a-13z7 + 2a-13z9 - 2a-12 + 11a-12z2 - 13a-12z4 + 5a-12z6 - 3a-12z8 + a-12z10 + 2a-11z - 11a-11z3 + 20a-11z5 - 14a-11z7 + 3a-11z9 + 3a-10z2 - 6a-10z4 + 8a-10z6 - 5a-10z8 + a-10z10 - 2a-9z + a-9z3 + 6a-9z5 - 5a-9z7 + a-9z9 + 3a-8 - 13a-8z2 + 16a-8z4 - 7a-8z6 + a-8z8

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

{12, 40}

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=8 is the signature of 11334. 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 = 31                       1 j = 29                     1   j = 27                   3 1   j = 25                 3 1     j = 23               4 3       j = 21             3 3         j = 19           4 4           j = 17         2 3             j = 15       2 4               j = 13     1 2                 j = 11     2                   j = 9 1 1                     j = 7 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, 334]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 334]]
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[2, 14, 3, 13], X[4, 16, 5, 15], > X[12, 18, 13, 17], X[8, 20, 9, 19], X[10, 22, 11, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 334]]
Out[4]=	GaussCode[1, -7, 2, -8, 3, -1, 4, -10, 5, -11, 6, -9, 7, -2, 8, -3, 9, -4, 10, > -5, 11, -6]
In[5]:=	DTCode[Knot[11, Alternating, 334]]
Out[5]=	DTCode[6, 14, 16, 18, 20, 22, 2, 4, 12, 8, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 334]][t]
Out[6]=	2 4 6 8 2 3 4 9 + -- - -- + -- - - - 8 t + 6 t - 4 t + 2 t 4 3 2 t t t t
In[7]:=	Conway[Knot[11, Alternating, 334]][z]
Out[7]=	2 4 6 8 1 + 12 z + 22 z + 12 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 334]}
In[9]:=	{KnotDet[Knot[11, Alternating, 334]], KnotSignature[Knot[11, Alternating, 334]]}
Out[9]=	{49, 8}
In[10]:=	J=Jones[Knot[11, Alternating, 334]][q]
Out[10]=	4 5 6 7 8 9 10 11 12 13 14 q - q + 3 q - 4 q + 6 q - 7 q + 7 q - 7 q + 6 q - 4 q + 2 q - 15 > q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 334]}
In[12]:=	A2Invariant[Knot[11, Alternating, 334]][q]
Out[12]=	14 18 22 24 28 30 32 34 40 44 q + 2 q + 2 q + q + q - 2 q + q - q - q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 334]][a, z]
Out[13]=	2 2 2 4 4 4 6 6 6 -2 3 7 z 6 z 13 z 5 z 11 z 16 z z 6 z 7 z --- + -- - ---- + ---- + ----- - ---- + ----- + ----- - --- + ---- + ---- + 12 8 12 10 8 12 10 8 12 10 8 a a a a a a a a a a a 8 8 z z > --- + -- 10 8 a a
In[14]:=	Kauffman[Knot[11, Alternating, 334]][a, z]
Out[14]=	2 2 2 2 -2 3 z z z z 2 z 2 z z 4 z 11 z 3 z --- + -- - --- + --- - --- + --- + --- - --- - --- + ---- + ----- + ---- - 12 8 19 17 15 13 11 9 18 16 12 10 a a a a a a a a a a a a 2 3 3 3 3 3 3 4 4 4 4 13 z z 2 z 5 z 4 z 11 z z 2 z 6 z z 13 z > ----- + --- - ---- + ---- - ---- - ----- + -- + ---- - ---- + --- - ----- - 8 19 17 15 13 11 9 18 16 14 12 a a a a a a a a a a a 4 4 5 5 5 5 5 6 6 6 6 z 16 z 3 z 8 z 3 z 20 z 6 z 4 z 6 z 5 z > ---- + ----- + ---- - ---- + ---- + ----- + ---- + ---- - ---- + ---- + 10 8 17 15 13 11 9 16 14 12 a a a a a a a a a a 6 6 7 7 7 7 8 8 8 8 9 8 z 7 z 4 z 5 z 14 z 5 z 3 z 3 z 5 z z 2 z > ---- - ---- + ---- - ---- - ----- - ---- + ---- - ---- - ---- + -- + ---- + 10 8 15 13 11 9 14 12 10 8 13 a a a a a a a a a a a 9 9 10 10 3 z z z z > ---- + -- + --- + --- 11 9 12 10 a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 334]], Vassiliev[3][Knot[11, Alternating, 334]]}
Out[15]=	{12, 40}
In[16]:=	Kh[Knot[11, Alternating, 334]][q, t]
Out[16]=	7 9 9 11 2 13 2 13 3 15 3 15 4 q + q + q t + 2 q t + q t + 2 q t + 2 q t + 4 q t + 17 4 17 5 19 5 19 6 21 6 21 7 > 2 q t + 3 q t + 4 q t + 4 q t + 3 q t + 3 q t + 23 7 23 8 25 8 25 9 27 9 27 10 29 10 > 4 q t + 3 q t + 3 q t + q t + 3 q t + q t + q t + 31 11 > q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a334

K11a333 K11a335