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

K11n77 K11n79

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

t-4 - 3t-3 + 6t-2 - 8t-1 + 9 - 8t + 6t2 - 3t3 + t4

Conway Polynomial:

1 + 5z2 + 8z4 + 5z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{1062, K11n76, ...}

Determinant and Signature:

{45, 4}

Jones Polynomial:

- q-1 + 2 - 4q + 7q2 - 6q3 + 8q4 - 7q5 + 5q6 - 4q7 + q8

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

{K11n76, ...}

A2 (sl(3)) Invariant:

- q-2 - 2q2 + q6 + 2q8 + 6q10 + 2q12 + 4q14 - 2q16 - 3q18 - 3q20 - 3q22 + q24 + q28

HOMFLY-PT Polynomial:

2a-8 + a-8z2 - 10a-6 - 13a-6z2 - 6a-6z4 - a-6z6 + 13a-4 + 25a-4z2 + 19a-4z4 + 7a-4z6 + a-4z8 - 4a-2 - 8a-2z2 - 5a-2z4 - a-2z6

Kauffman Polynomial:

a-10z2 - 2a-9z + 4a-9z3 + 2a-8 + a-8z2 - 2a-8z4 + 2a-8z6 - 14a-7z + 26a-7z3 - 18a-7z5 + 5a-7z7 + 10a-6 - 17a-6z2 + 16a-6z4 - 13a-6z6 + 4a-6z8 - 21a-5z + 42a-5z3 - 31a-5z5 + 5a-5z7 + a-5z9 + 13a-4 - 26a-4z2 + 31a-4z4 - 24a-4z6 + 6a-4z8 - 13a-3z + 28a-3z3 - 18a-3z5 + a-3z7 + a-3z9 + 4a-2 - 9a-2z2 + 13a-2z4 - 9a-2z6 + 2a-2z8 - 4a-1z + 8a-1z3 - 5a-1z5 + a-1z7

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

{5, 7}

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 1178. 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 r = 6 j = 17                   1 j = 15                 3   j = 13               2 1   j = 11             5 3     j = 9           3 2       j = 7         3 5         j = 5       4 3           j = 3     1 4             j = 1   1 3               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, 78]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 78]]
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, 20, 10, 21], X[11, 17, 12, 16], X[13, 19, 14, 18], X[15, 7, 16, 6], > X[17, 13, 18, 12], X[19, 22, 20, 1], X[21, 10, 22, 11]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 78]]
Out[4]=	GaussCode[1, -4, 2, -1, -3, 8, 4, -2, -5, 11, -6, 9, -7, 3, -8, 6, -9, 7, -10, > 5, -11, 10]
In[5]:=	DTCode[Knot[11, NonAlternating, 78]]
Out[5]=	DTCode[4, 8, -14, 2, -20, -16, -18, -6, -12, -22, -10]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 78]][t]
Out[6]=	-4 3 6 8 2 3 4 9 + t - -- + -- - - - 8 t + 6 t - 3 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 78]][z]
Out[7]=	2 4 6 8 1 + 5 z + 8 z + 5 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 62], Knot[11, NonAlternating, 76], Knot[11, NonAlternating, 78]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 78]], KnotSignature[Knot[11, NonAlternating, 78]]}
Out[9]=	{45, 4}
In[10]:=	J=Jones[Knot[11, NonAlternating, 78]][q]
Out[10]=	1 2 3 4 5 6 7 8 2 - - - 4 q + 7 q - 6 q + 8 q - 7 q + 5 q - 4 q + q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 76], Knot[11, NonAlternating, 78]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 78]][q]
Out[12]=	-2 2 6 8 10 12 14 16 18 20 -q - 2 q + q + 2 q + 6 q + 2 q + 4 q - 2 q - 3 q - 3 q - 22 24 28 > 3 q + q + q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 78]][a, z]
Out[13]=	2 2 2 2 4 4 4 6 2 10 13 4 z 13 z 25 z 8 z 6 z 19 z 5 z z -- - -- + -- - -- + -- - ----- + ----- - ---- - ---- + ----- - ---- - -- + 8 6 4 2 8 6 4 2 6 4 2 6 a 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, 78]][a, z]
Out[14]=	2 2 2 2 2 10 13 4 2 z 14 z 21 z 13 z 4 z z z 17 z 26 z -- + -- + -- + -- - --- - ---- - ---- - ---- - --- + --- + -- - ----- - ----- - 8 6 4 2 9 7 5 3 a 10 8 6 4 a a a a a a a a a a a a 2 3 3 3 3 3 4 4 4 4 9 z 4 z 26 z 42 z 28 z 8 z 2 z 16 z 31 z 13 z > ---- + ---- + ----- + ----- + ----- + ---- - ---- + ----- + ----- + ----- - 2 9 7 5 3 a 8 6 4 2 a a a a a a a a a 5 5 5 5 6 6 6 6 7 7 18 z 31 z 18 z 5 z 2 z 13 z 24 z 9 z 5 z 5 z > ----- - ----- - ----- - ---- + ---- - ----- - ----- - ---- + ---- + ---- + 7 5 3 a 8 6 4 2 7 5 a a a a a a a a a 7 7 8 8 8 9 9 z z 4 z 6 z 2 z z z > -- + -- + ---- + ---- + ---- + -- + -- 3 a 6 4 2 5 3 a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 78]], Vassiliev[3][Knot[11, NonAlternating, 78]]}
Out[15]=	{5, 7}
In[16]:=	Kh[Knot[11, NonAlternating, 78]][q, t]
Out[16]=	3 3 5 1 1 q 3 q q 5 7 7 2 4 q + 4 q + ----- + ---- + -- + --- + -- + 3 q t + 3 q t + 5 q t + 3 3 2 2 t t q t q t t 9 2 9 3 11 3 11 4 13 4 13 5 15 5 > 3 q t + 2 q t + 5 q t + 3 q t + 2 q t + q t + 3 q t + 17 6 > q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n78

K11n77 K11n79