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

K11n60 K11n62

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

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

Conway Polynomial:

1 + 4z2 + 6z4 + 5z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{17, 4}

Jones Polynomial:

- q-1 + 2 - 2q + 3q2 - 2q3 + 3q4 - 2q5 + q6 - q7

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

{10130, ...}

A2 (sl(3)) Invariant:

- q-2 + q6 + 2q8 + 3q10 + q12 + 2q14 - q16 - q18 - 2q20 - q22 - q26 + q28

HOMFLY-PT Polynomial:

a-8 + a-8z2 - 6a-6 - 10a-6z2 - 6a-6z4 - a-6z6 + 8a-4 + 19a-4z2 + 17a-4z4 + 7a-4z6 + a-4z8 - 2a-2 - 6a-2z2 - 5a-2z4 - a-2z6

Kauffman Polynomial:

a-8 - a-8z2 - 6a-7z + 10a-7z3 - 6a-7z5 + a-7z7 + 6a-6 - 18a-6z2 + 22a-6z4 - 12a-6z6 + 2a-6z8 - 10a-5z + 18a-5z3 - 6a-5z5 - 3a-5z7 + a-5z9 + 8a-4 - 27a-4z2 + 39a-4z4 - 23a-4z6 + 4a-4z8 - 6a-3z + 14a-3z3 - 5a-3z5 - 3a-3z7 + a-3z9 + 2a-2 - 10a-2z2 + 17a-2z4 - 11a-2z6 + 2a-2z8 - 2a-1z + 6a-1z3 - 5a-1z5 + a-1z7

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

{4, 6}

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 1161. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 15                 1 j = 13               1 1 j = 11             2 1   j = 9           1 1 1   j = 7         2 3       j = 5       2 1         j = 3     1 3 1         j = 1   1 1             j = -1   1               j = -3 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, 61]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 61]]
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, 1, 14, 22], X[15, 7, 16, 6], > X[17, 20, 18, 21], X[19, 12, 20, 13], X[21, 11, 22, 10]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 61]]
Out[4]=	GaussCode[1, -4, 2, -1, -3, 8, 4, -2, -5, 11, -6, 10, -7, 3, -8, 5, -9, 6, -10, > 9, -11, 7]
In[5]:=	DTCode[Knot[11, NonAlternating, 61]]
Out[5]=	DTCode[4, 8, -14, 2, -16, -18, -22, -6, -20, -12, -10]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 61]][t]
Out[6]=	-4 3 4 1 2 3 4 -1 + t - -- + -- - - - t + 4 t - 3 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 61]][z]
Out[7]=	2 4 6 8 1 + 4 z + 6 z + 5 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 61]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 61]], KnotSignature[Knot[11, NonAlternating, 61]]}
Out[9]=	{17, 4}
In[10]:=	J=Jones[Knot[11, NonAlternating, 61]][q]
Out[10]=	1 2 3 4 5 6 7 2 - - - 2 q + 3 q - 2 q + 3 q - 2 q + q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 130], Knot[11, NonAlternating, 61]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 61]][q]
Out[12]=	-2 6 8 10 12 14 16 18 20 22 26 28 -q + q + 2 q + 3 q + q + 2 q - q - q - 2 q - q - q + q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 61]][a, z]
Out[13]=	2 2 2 2 4 4 4 6 -8 6 8 2 z 10 z 19 z 6 z 6 z 17 z 5 z z a - -- + -- - -- + -- - ----- + ----- - ---- - ---- + ----- - ---- - -- + 6 4 2 8 6 4 2 6 4 2 6 a a a a a a a a a a a 6 6 8 7 z z z > ---- - -- + -- 4 2 4 a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 61]][a, z]
Out[14]=	2 2 2 2 -8 6 8 2 6 z 10 z 6 z 2 z z 18 z 27 z 10 z a + -- + -- + -- - --- - ---- - --- - --- - -- - ----- - ----- - ----- + 6 4 2 7 5 3 a 8 6 4 2 a a a a a a a a a a 3 3 3 3 4 4 4 5 5 5 10 z 18 z 14 z 6 z 22 z 39 z 17 z 6 z 6 z 5 z > ----- + ----- + ----- + ---- + ----- + ----- + ----- - ---- - ---- - ---- - 7 5 3 a 6 4 2 7 5 3 a a a a a a a a a 5 6 6 6 7 7 7 7 8 8 8 5 z 12 z 23 z 11 z z 3 z 3 z z 2 z 4 z 2 z > ---- - ----- - ----- - ----- + -- - ---- - ---- + -- + ---- + ---- + ---- + a 6 4 2 7 5 3 a 6 4 2 a a a a a a a a a 9 9 z z > -- + -- 5 3 a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 61]], Vassiliev[3][Knot[11, NonAlternating, 61]]}
Out[15]=	{4, 6}
In[16]:=	Kh[Knot[11, NonAlternating, 61]][q, t]
Out[16]=	3 3 5 1 1 q q q 3 5 7 7 2 3 q + 2 q + ----- + ---- + -- + - + -- + q t + q t + 2 q t + 3 q t + 3 3 2 2 t t q t q t t 9 2 9 3 11 3 9 4 11 4 13 4 13 5 15 5 > q t + q t + 2 q t + q t + q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n61

K11n60 K11n62