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

K11a30 K11a32

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X14,6,15,5 X2837 X16,10,17,9 X18,11,19,12 X6,14,7,13 X22,16,1,15 X20,18,21,17 X10,19,11,20 X12,22,13,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 14 2 16 18 6 22 20 10 12

Alexander Polynomial:

- 3t-3 + 14t-2 - 28t-1 + 35 - 28t + 14t2 - 3t3

Conway Polynomial:

1 + z2 - 4z4 - 3z6

Other knots with the same Alexander/Conway Polynomial:

{K11a317, ...}

Determinant and Signature:

{125, 4}

Jones Polynomial:

1 - 3q + 8q2 - 12q3 + 17q4 - 20q5 + 20q6 - 18q7 + 13q8 - 8q9 + 4q10 - q11

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

{...}

A2 (sl(3)) Invariant:

1 - q2 + q4 + 3q6 - 2q8 + 5q10 - q12 - q14 + q16 - 5q18 + 2q20 - 3q22 + q24 + 3q26 - 2q28 + 2q30 - q34

HOMFLY-PT Polynomial:

- a-10 - a-10z2 + 4a-8 + 7a-8z2 + 3a-8z4 - 6a-6 - 10a-6z2 - 7a-6z4 - 2a-6z6 + 3a-4 + 3a-4z2 - a-4z4 - a-4z6 + a-2 + 2a-2z2 + a-2z4

Kauffman Polynomial:

- a-13z3 + a-13z5 + 2a-12z2 - 6a-12z4 + 4a-12z6 - 2a-11z + 6a-11z3 - 11a-11z5 + 7a-11z7 + a-10 + 2a-10z2 - 3a-10z4 - 6a-10z6 + 7a-10z8 - 7a-9z + 26a-9z3 - 29a-9z5 + 8a-9z7 + 4a-9z9 + 4a-8 - 13a-8z2 + 30a-8z4 - 35a-8z6 + 15a-8z8 + a-8z10 - 9a-7z + 25a-7z3 - 20a-7z5 - 4a-7z7 + 8a-7z9 + 6a-6 - 22a-6z2 + 38a-6z4 - 37a-6z6 + 13a-6z8 + a-6z10 - 4a-5z + 10a-5z3 - 10a-5z5 - 2a-5z7 + 4a-5z9 + 3a-4 - 6a-4z2 + 8a-4z4 - 11a-4z6 + 5a-4z8 + 4a-3z3 - 7a-3z5 + 3a-3z7 - a-2 + 3a-2z2 - 3a-2z4 + a-2z6

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=4 is the signature of 1131. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 j = 23                       1 j = 21                     3   j = 19                   5 1   j = 17                 8 3     j = 15               10 5       j = 13             10 8         j = 11           10 10           j = 9         7 10             j = 7       5 10               j = 5     3 7                 j = 3   1 6                   j = 1   2                     j = -1 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, 31]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 31]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[14, 6, 15, 5], X[2, 8, 3, 7], > X[16, 10, 17, 9], X[18, 11, 19, 12], X[6, 14, 7, 13], X[22, 16, 1, 15], > X[20, 18, 21, 17], X[10, 19, 11, 20], X[12, 22, 13, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 31]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -7, 4, -2, 5, -10, 6, -11, 7, -3, 8, -5, 9, -6, 10, > -9, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 31]]
Out[5]=	DTCode[4, 8, 14, 2, 16, 18, 6, 22, 20, 10, 12]
In[6]:=	alex = Alexander[Knot[11, Alternating, 31]][t]
Out[6]=	3 14 28 2 3 35 - -- + -- - -- - 28 t + 14 t - 3 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 31]][z]
Out[7]=	2 4 6 1 + z - 4 z - 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 31], Knot[11, Alternating, 317]}
In[9]:=	{KnotDet[Knot[11, Alternating, 31]], KnotSignature[Knot[11, Alternating, 31]]}
Out[9]=	{125, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 31]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 1 - 3 q + 8 q - 12 q + 17 q - 20 q + 20 q - 18 q + 13 q - 8 q + 4 q - 11 > q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 31]}
In[12]:=	A2Invariant[Knot[11, Alternating, 31]][q]
Out[12]=	2 4 6 8 10 12 14 16 18 20 22 1 - q + q + 3 q - 2 q + 5 q - q - q + q - 5 q + 2 q - 3 q + 24 26 28 30 34 > q + 3 q - 2 q + 2 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 31]][a, z]
Out[13]=	2 2 2 2 2 4 4 -10 4 6 3 -2 z 7 z 10 z 3 z 2 z 3 z 7 z -a + -- - -- + -- + a - --- + ---- - ----- + ---- + ---- + ---- - ---- - 8 6 4 10 8 6 4 2 8 6 a a a a a a a a a a 4 4 6 6 z z 2 z z > -- + -- - ---- - -- 4 2 6 4 a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 31]][a, z]
Out[14]=	2 2 2 -10 4 6 3 -2 2 z 7 z 9 z 4 z 2 z 2 z 13 z a + -- + -- + -- - a - --- - --- - --- - --- + ---- + ---- - ----- - 8 6 4 11 9 7 5 12 10 8 a a a a a a a a a a 2 2 2 3 3 3 3 3 3 4 22 z 6 z 3 z z 6 z 26 z 25 z 10 z 4 z 6 z > ----- - ---- + ---- - --- + ---- + ----- + ----- + ----- + ---- - ---- - 6 4 2 13 11 9 7 5 3 12 a a a a a a a a a a 4 4 4 4 4 5 5 5 5 5 3 z 30 z 38 z 8 z 3 z z 11 z 29 z 20 z 10 z > ---- + ----- + ----- + ---- - ---- + --- - ----- - ----- - ----- - ----- - 10 8 6 4 2 13 11 9 7 5 a a a a a a a a a a 5 6 6 6 6 6 6 7 7 7 7 z 4 z 6 z 35 z 37 z 11 z z 7 z 8 z 4 z > ---- + ---- - ---- - ----- - ----- - ----- + -- + ---- + ---- - ---- - 3 12 10 8 6 4 2 11 9 7 a a a a a a a a a a 7 7 8 8 8 8 9 9 9 10 10 2 z 3 z 7 z 15 z 13 z 5 z 4 z 8 z 4 z z z > ---- + ---- + ---- + ----- + ----- + ---- + ---- + ---- + ---- + --- + --- 5 3 10 8 6 4 9 7 5 8 6 a a a a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 31]], Vassiliev[3][Knot[11, Alternating, 31]]}
Out[15]=	{1, 1}
In[16]:=	Kh[Knot[11, Alternating, 31]][q, t]
Out[16]=	3 3 5 1 2 q q 5 7 7 2 9 2 6 q + 3 q + ---- + --- + -- + 7 q t + 5 q t + 10 q t + 7 q t + 2 t t q t 9 3 11 3 11 4 13 4 13 5 15 5 > 10 q t + 10 q t + 10 q t + 10 q t + 8 q t + 10 q t + 15 6 17 6 17 7 19 7 19 8 21 8 23 9 > 5 q t + 8 q t + 3 q t + 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a31

K11a30 K11a32