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

K11n30 K11n32

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X12,6,13,5 X2837 X9,17,10,16 X6,12,7,11 X13,20,14,21 X15,11,16,10 X17,1,18,22 X19,14,20,15 X21,19,22,18

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 12 2 -16 6 -20 -10 -22 -14 -18

Alexander Polynomial:

- t-3 + 2t-2 + 2t-1 - 5 + 2t + 2t2 - t3

Conway Polynomial:

1 + z2 - 4z4 - z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{3, 2}

Jones Polynomial:

1 - q + 2q2 - q3 - q6 + 2q7 - 2q8 + 2q9 - q10

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

{...}

A2 (sl(3)) Invariant:

1 + q2 + q4 + q6 + q8 - 2q12 - 2q16 + q22 + 2q24 + q28 - q30 - q32

HOMFLY-PT Polynomial:

- a-10 + 2a-8 + 2a-8z2 + a-6z2 - 3a-4 - 6a-4z2 - 5a-4z4 - a-4z6 + 3a-2 + 4a-2z2 + a-2z4

Kauffman Polynomial:

- 2a-11z + 6a-11z3 - 5a-11z5 + a-11z7 + a-10 - 8a-10z2 + 16a-10z4 - 11a-10z6 + 2a-10z8 - 2a-9z + 8a-9z3 - a-9z5 - 4a-9z7 + a-9z9 + 2a-8 - 17a-8z2 + 32a-8z4 - 19a-8z6 + 3a-8z8 + 2a-7z - 4a-7z3 + 8a-7z5 - 6a-7z7 + a-7z9 - 6a-6z2 + 11a-6z4 - 7a-6z6 + a-6z8 + 3a-5z - 4a-5z3 - 3a-4 + 10a-4z2 - 10a-4z4 + 2a-4z6 + a-3z + 2a-3z3 - 4a-3z5 + a-3z7 - 3a-2 + 7a-2z2 - 5a-2z4 + a-2z6

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

{1, 3}

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

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n31

K11n30 K11n32