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

K11n59 K11n61

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X5,15,6,14 X2837 X9,17,10,16 X11,18,12,19 X13,20,14,21 X15,7,16,6 X17,1,18,22 X19,12,20,13 X21,10,22,11

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- t-4 + 3t-3 - 4t-2 + 5t-1 - 5 + 5t - 4t2 + 3t3 - t4

Conway Polynomial:

1 - 6z4 - 5z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{1046, ...}

Determinant and Signature:

{31, 2}

Jones Polynomial:

q-3 - 2q-2 + 3q-1 - 4 + 5q - 5q2 + 5q3 - 3q4 + 2q5 - q6

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

{98, ...}

A2 (sl(3)) Invariant:

q-8 + q-4 - 2q4 + q6 - q8 + 2q10 + q12 + q14 + q16 - q18 - q22

HOMFLY-PT Polynomial:

- 2a-6 - a-6z2 + 6a-4 + 11a-4z2 + 6a-4z4 + a-4z6 - 6a-2 - 17a-2z2 - 17a-2z4 - 7a-2z6 - a-2z8 + 3 + 7z2 + 5z4 + z6

Kauffman Polynomial:

- 2a-7z + a-7z3 + 2a-6 - 4a-6z2 + 2a-6z4 - 4a-5z + 8a-5z3 - 4a-5z5 + a-5z7 + 6a-4 - 19a-4z2 + 24a-4z4 - 11a-4z6 + 2a-4z8 - 4a-3z + 8a-3z3 - 3a-3z7 + a-3z9 + 6a-2 - 25a-2z2 + 35a-2z4 - 20a-2z6 + 4a-2z8 - 4a-1z + 8a-1z3 - 4a-1z5 - 2a-1z7 + a-1z9 + 3 - 7z2 + 9z4 - 8z6 + 2z8 - 2az + 7az3 - 8az5 + 2az7 + 3a2z2 - 4a2z4 + a2z6

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

{0, 2}

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 1160. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 13                   1 j = 11                 1   j = 9               2 1   j = 7             3 1     j = 5           2 2       j = 3         3 3         j = 1       2 3           j = -1     1 2             j = -3   1 2               j = -5   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, NonAlternating, 60]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 60]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 15, 6, 14], X[2, 8, 3, 7], > X[9, 17, 10, 16], X[11, 18, 12, 19], X[13, 20, 14, 21], X[15, 7, 16, 6], > X[17, 1, 18, 22], X[19, 12, 20, 13], X[21, 10, 22, 11]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 60]]
Out[4]=	GaussCode[1, -4, 2, -1, -3, 8, 4, -2, -5, 11, -6, 10, -7, 3, -8, 5, -9, 6, -10, > 7, -11, 9]
In[5]:=	DTCode[Knot[11, NonAlternating, 60]]
Out[5]=	DTCode[4, 8, -14, 2, -16, -18, -20, -6, -22, -12, -10]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 60]][t]
Out[6]=	-4 3 4 5 2 3 4 -5 - t + -- - -- + - + 5 t - 4 t + 3 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 60]][z]
Out[7]=	4 6 8 1 - 6 z - 5 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 46], Knot[11, NonAlternating, 60]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 60]], KnotSignature[Knot[11, NonAlternating, 60]]}
Out[9]=	{31, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 60]][q]
Out[10]=	-3 2 3 2 3 4 5 6 -4 + q - -- + - + 5 q - 5 q + 5 q - 3 q + 2 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 8], Knot[11, NonAlternating, 60]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 60]][q]
Out[12]=	-8 -4 4 6 8 10 12 14 16 18 22 q + q - 2 q + q - q + 2 q + q + q + q - q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 60]][a, z]
Out[13]=	2 2 2 4 4 6 2 6 6 2 z 11 z 17 z 4 6 z 17 z 6 z 3 - -- + -- - -- + 7 z - -- + ----- - ----- + 5 z + ---- - ----- + z + -- - 6 4 2 6 4 2 4 2 4 a a a a a a a a a 6 8 7 z z > ---- - -- 2 2 a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 60]][a, z]
Out[14]=	2 2 2 6 6 2 z 4 z 4 z 4 z 2 4 z 19 z 3 + -- + -- + -- - --- - --- - --- - --- - 2 a z - 7 z - ---- - ----- - 6 4 2 7 5 3 a 6 4 a a a a a a a a 2 3 3 3 3 4 4 25 z 2 2 z 8 z 8 z 8 z 3 4 2 z 24 z > ----- + 3 a z + -- + ---- + ---- + ---- + 7 a z + 9 z + ---- + ----- + 2 7 5 3 a 6 4 a a a a a a 4 5 5 6 6 35 z 2 4 4 z 4 z 5 6 11 z 20 z 2 6 > ----- - 4 a z - ---- - ---- - 8 a z - 8 z - ----- - ----- + a z + 2 5 a 4 2 a a a a 7 7 7 8 8 9 9 z 3 z 2 z 7 8 2 z 4 z z z > -- - ---- - ---- + 2 a z + 2 z + ---- + ---- + -- + -- 5 3 a 4 2 3 a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 60]], Vassiliev[3][Knot[11, NonAlternating, 60]]}
Out[15]=	{0, 2}
In[16]:=	Kh[Knot[11, NonAlternating, 60]][q, t]
Out[16]=	3 1 1 1 2 1 2 2 q 3 3 q + 3 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 3 q t + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 13 5 > 2 q t + 2 q t + 3 q t + q t + 2 q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n60

K11n59 K11n61