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

K11n134 K11n136

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X12,3,13,4 X14,6,15,5 X7,17,8,16 X22,10,1,9 X11,19,12,18 X2,13,3,14 X15,20,16,21 X17,11,18,10 X19,7,20,6 X8,22,9,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

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

Conway Polynomial:

1 - z2 - 4z4 - z6

Other knots with the same Alexander/Conway Polynomial:

{K11n19, ...}

Determinant and Signature:

{5, 4}

Jones Polynomial:

1 - q + 2q2 - q3 + q4 - q5 - q8 + q9

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

{...}

A2 (sl(3)) Invariant:

1 + q2 + q4 + q6 + q8 + q10 - q12 - 2q16 - q18 - q20 + q24 + q28

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

5a-10z2 - 5a-10z4 + a-10z6 - 3a-9z + 9a-9z3 - 6a-9z5 + a-9z7 + a-8 + a-8z2 - a-8z4 - 3a-7z + 3a-7z3 - a-7z5 + a-6 - 2a-6z2 + 2a-5z - 4a-5z3 + a-5z5 - 2a-4 + 9a-4z2 - 9a-4z4 + 2a-4z6 + 2a-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, -4}

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 11135. 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 j = 19                     1 j = 17                       j = 15               1 1 1   j = 13             1 1       j = 11           1 1 1       j = 9         1 2 1         j = 7       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, 135]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 135]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 3, 13, 4], X[14, 6, 15, 5], X[7, 17, 8, 16], > X[22, 10, 1, 9], X[11, 19, 12, 18], X[2, 13, 3, 14], X[15, 20, 16, 21], > X[17, 11, 18, 10], X[19, 7, 20, 6], X[8, 22, 9, 21]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 135]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, 10, -4, -11, 5, 9, -6, -2, 7, -3, -8, 4, -9, 6, -10, > 8, 11, -5]
In[5]:=	DTCode[Knot[11, NonAlternating, 135]]
Out[5]=	DTCode[4, 12, 14, -16, 22, -18, 2, -20, -10, -6, 8]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 135]][t]
Out[6]=	-3 2 2 3 -1 - t + -- + 2 t - t 2 t
In[7]:=	Conway[Knot[11, NonAlternating, 135]][z]
Out[7]=	2 4 6 1 - z - 4 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 19], Knot[11, NonAlternating, 135]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 135]], KnotSignature[Knot[11, NonAlternating, 135]]}
Out[9]=	{5, 4}
In[10]:=	J=Jones[Knot[11, NonAlternating, 135]][q]
Out[10]=	2 3 4 5 8 9 1 - q + 2 q - q + q - q - q + q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 135]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 135]][q]
Out[12]=	2 4 6 8 10 12 16 18 20 24 28 1 + q + q + q + q + q - q - 2 q - q - q + q + q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 135]][a, z]
Out[13]=	2 2 2 4 4 6 -8 -6 2 3 z 6 z 4 z 5 z z z a - a - -- + -- + -- - ---- + ---- - ---- + -- - -- 4 2 8 4 2 4 2 4 a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 135]][a, z]
Out[14]=	2 2 2 2 2 -8 -6 2 3 3 z 3 z 2 z 2 z 5 z z 2 z 9 z 7 z a + a - -- - -- - --- - --- + --- + --- + ---- + -- - ---- + ---- + ---- + 4 2 9 7 5 3 10 8 6 4 2 a a a a a a a a a a a 3 3 3 3 4 4 4 4 5 5 5 9 z 3 z 4 z 2 z 5 z z 9 z 5 z 6 z z z > ---- + ---- - ---- + ---- - ---- - -- - ---- - ---- - ---- - -- + -- - 9 7 5 3 10 8 4 2 9 7 5 a a a a a a a a a a a 5 6 6 6 7 7 4 z z 2 z z z z > ---- + --- + ---- + -- + -- + -- 3 10 4 2 9 3 a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 135]], Vassiliev[3][Knot[11, NonAlternating, 135]]}
Out[15]=	{-1, -4}
In[16]:=	Kh[Knot[11, NonAlternating, 135]][q, t]
Out[16]=	3 3 5 1 q 5 7 5 2 7 2 9 2 9 3 2 q + q + ---- + -- + 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 15 6 > q t + q t + q t + q t + q t + q t + q t + q t + 15 7 19 8 > q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n135

K11n134 K11n136