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

K11n135 K11n137

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X12,4,13,3 X5,17,6,16 X14,8,15,7 X18,10,19,9 X2,12,3,11 X8,14,9,13 X15,1,16,22 X20,18,21,17 X10,20,11,19 X21,7,22,6

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

3t-3 - 8t-2 + 13t-1 - 15 + 13t - 8t2 + 3t3

Conway Polynomial:

1 + 8z2 + 10z4 + 3z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{63, 6}

Jones Polynomial:

q3 - 2q4 + 6q5 - 8q6 + 10q7 - 11q8 + 10q9 - 8q10 + 5q11 - 2q12

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

{...}

A2 (sl(3)) Invariant:

q10 - q12 + 3q14 + 2q18 + 2q20 - 2q22 + 2q24 - 3q26 + q28 - q32 + q34 - 2q36

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

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

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

{8, 22}

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=6 is the signature of 11136. 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 j = 25                   2 j = 23                 3   j = 21               5 2   j = 19             5 3     j = 17           6 5       j = 15         4 5         j = 13       4 6           j = 11     2 4             j = 9     4               j = 7 1 2                 j = 5 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, 136]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 136]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 4, 13, 3], X[5, 17, 6, 16], X[14, 8, 15, 7], > X[18, 10, 19, 9], X[2, 12, 3, 11], X[8, 14, 9, 13], X[15, 1, 16, 22], > X[20, 18, 21, 17], X[10, 20, 11, 19], X[21, 7, 22, 6]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 136]]
Out[4]=	GaussCode[1, -6, 2, -1, -3, 11, 4, -7, 5, -10, 6, -2, 7, -4, -8, 3, 9, -5, 10, > -9, -11, 8]
In[5]:=	DTCode[Knot[11, NonAlternating, 136]]
Out[5]=	DTCode[4, 12, -16, 14, 18, 2, 8, -22, 20, 10, -6]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 136]][t]
Out[6]=	3 8 13 2 3 -15 + -- - -- + -- + 13 t - 8 t + 3 t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 136]][z]
Out[7]=	2 4 6 1 + 8 z + 10 z + 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 136]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 136]], KnotSignature[Knot[11, NonAlternating, 136]]}
Out[9]=	{63, 6}
In[10]:=	J=Jones[Knot[11, NonAlternating, 136]][q]
Out[10]=	3 4 5 6 7 8 9 10 11 12 q - 2 q + 6 q - 8 q + 10 q - 11 q + 10 q - 8 q + 5 q - 2 q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 136]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 136]][q]
Out[12]=	10 12 14 18 20 22 24 26 28 32 34 q - q + 3 q + 2 q + 2 q - 2 q + 2 q - 3 q + q - q + q - 36 > 2 q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 136]][a, z]
Out[13]=	2 2 2 4 4 4 6 6 -2 -8 2 5 z 8 z 5 z 2 z 8 z 4 z 2 z z --- + a + -- - ---- + ---- + ---- - ---- + ---- + ---- + ---- + -- 10 6 10 8 6 10 8 6 8 6 a a a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 136]][a, z]
Out[14]=	2 2 2 2 2 2 -8 2 2 z z 3 z 3 z z 3 z z 6 z 5 z 5 z --- + a - -- - --- - --- - --- - --- + -- - ---- + --- - ---- - ---- + ---- + 10 6 15 13 11 9 7 14 12 10 8 6 a a a a a a a a a a a a 3 3 3 3 3 4 4 4 4 5 3 z 3 z 12 z 14 z 2 z 4 z 7 z 7 z 4 z 2 z > ---- + ---- + ----- + ----- + ---- + ---- + ---- + ---- - ---- - ---- - 15 13 11 9 7 14 10 8 6 13 a a a a a a a a a a 5 5 5 6 6 6 6 6 7 7 16 z 19 z 5 z z 2 z 12 z 8 z z 3 z 6 z > ----- - ----- - ---- + --- - ---- - ----- - ---- + -- + ---- + ---- + 11 9 7 14 12 10 8 6 13 11 a a a a a a a a a a 7 7 8 8 8 9 9 5 z 2 z 3 z 6 z 3 z z z > ---- + ---- + ---- + ---- + ---- + --- + -- 9 7 12 10 8 11 9 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 136]], Vassiliev[3][Knot[11, NonAlternating, 136]]}
Out[15]=	{8, 22}
In[16]:=	Kh[Knot[11, NonAlternating, 136]][q, t]
Out[16]=	5 7 7 9 2 11 2 11 3 13 3 13 4 q + q + 2 q t + 4 q t + 2 q t + 4 q t + 4 q t + 6 q t + 15 4 15 5 17 5 17 6 19 6 19 7 > 4 q t + 5 q t + 6 q t + 5 q t + 5 q t + 3 q t + 21 7 21 8 23 8 25 9 > 5 q t + 2 q t + 3 q t + 2 q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n136

K11n135 K11n137