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

K11n76 K11n78

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X5,15,6,14 X2837 X20,10,21,9 X11,17,12,16 X13,19,14,18 X15,7,16,6 X17,13,18,12 X22,20,1,19 X10,22,11,21

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 - t-3 - 2t-2 + 8t-1 - 11 + 8t - 2t2 - t3 + t4

Conway Polynomial:

1 + 7z2 + 12z4 + 7z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{27, 6}

Jones Polynomial:

q4 + q6 + 2q7 - 4q8 + 4q9 - 6q10 + 5q11 - 4q12 + 3q13 - q14

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

{...}

A2 (sl(3)) Invariant:

q14 + q16 + 2q18 + 4q20 + 2q22 + q24 - 5q28 - 3q30 - 4q32 + 2q36 + q38 + 3q40 - q42 - q44

HOMFLY-PT Polynomial:

- a-14 + 6a-12 + 4a-12z2 - 13a-10 - 20a-10z2 - 9a-10z4 - a-10z6 + 9a-8 + 23a-8z2 + 21a-8z4 + 8a-8z6 + a-8z8

Kauffman Polynomial:

- 2a-17z3 + a-17z5 + 3a-16z2 - 8a-16z4 + 3a-16z6 - 3a-15z + 5a-15z3 - 8a-15z5 + 3a-15z7 + a-14 + 5a-14z2 - 8a-14z4 + a-14z6 + a-14z8 - 12a-13z + 23a-13z3 - 15a-13z5 + 4a-13z7 + 6a-12 - 6a-12z2 + 8a-12z4 - 4a-12z6 + a-12z8 - 22a-11z + 36a-11z3 - 15a-11z5 + 2a-11z7 + 13a-10 - 31a-10z2 + 29a-10z4 - 10a-10z6 + a-10z8 - 13a-9z + 20a-9z3 - 9a-9z5 + a-9z7 + 9a-8 - 23a-8z2 + 21a-8z4 - 8a-8z6 + a-8z8

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

{7, 16}

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 1177. 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 r = 10 r = 11 j = 29                       1 j = 27                     2   j = 25                   2 1   j = 23                 3 2     j = 21               3 2       j = 19           1 2 3         j = 17           3 3           j = 15       1 1 2             j = 13         3               j = 11     1                   j = 9 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, 77]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 77]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 15, 6, 14], X[2, 8, 3, 7], > X[20, 10, 21, 9], X[11, 17, 12, 16], X[13, 19, 14, 18], X[15, 7, 16, 6], > X[17, 13, 18, 12], X[22, 20, 1, 19], X[10, 22, 11, 21]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 77]]
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, 77]]
Out[5]=	DTCode[4, 8, -14, 2, 20, -16, -18, -6, -12, 22, 10]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 77]][t]
Out[6]=	-4 -3 2 8 2 3 4 -11 + t - t - -- + - + 8 t - 2 t - t + t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 77]][z]
Out[7]=	2 4 6 8 1 + 7 z + 12 z + 7 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 77]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 77]], KnotSignature[Knot[11, NonAlternating, 77]]}
Out[9]=	{27, 6}
In[10]:=	J=Jones[Knot[11, NonAlternating, 77]][q]
Out[10]=	4 6 7 8 9 10 11 12 13 14 q + q + 2 q - 4 q + 4 q - 6 q + 5 q - 4 q + 3 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 77]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 77]][q]
Out[12]=	14 16 18 20 22 24 28 30 32 36 38 q + q + 2 q + 4 q + 2 q + q - 5 q - 3 q - 4 q + 2 q + q + 40 42 44 > 3 q - q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 77]][a, z]
Out[13]=	2 2 2 4 4 6 6 8 -14 6 13 9 4 z 20 z 23 z 9 z 21 z z 8 z z -a + --- - --- + -- + ---- - ----- + ----- - ---- + ----- - --- + ---- + -- 12 10 8 12 10 8 10 8 10 8 8 a a a a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 77]][a, z]
Out[14]=	2 2 2 2 -14 6 13 9 3 z 12 z 22 z 13 z 3 z 5 z 6 z 31 z a + --- + --- + -- - --- - ---- - ---- - ---- + ---- + ---- - ---- - ----- - 12 10 8 15 13 11 9 16 14 12 10 a a a a a a a a a a a 2 3 3 3 3 3 4 4 4 4 23 z 2 z 5 z 23 z 36 z 20 z 8 z 8 z 8 z 29 z > ----- - ---- + ---- + ----- + ----- + ----- - ---- - ---- + ---- + ----- + 8 17 15 13 11 9 16 14 12 10 a a a a a a a a a a 4 5 5 5 5 5 6 6 6 6 21 z z 8 z 15 z 15 z 9 z 3 z z 4 z 10 z > ----- + --- - ---- - ----- - ----- - ---- + ---- + --- - ---- - ----- - 8 17 15 13 11 9 16 14 12 10 a a a a a a a a a a 6 7 7 7 7 8 8 8 8 8 z 3 z 4 z 2 z z z z z z > ---- + ---- + ---- + ---- + -- + --- + --- + --- + -- 8 15 13 11 9 14 12 10 8 a a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 77]], Vassiliev[3][Knot[11, NonAlternating, 77]]}
Out[15]=	{7, 16}
In[16]:=	Kh[Knot[11, NonAlternating, 77]][q, t]
Out[16]=	7 9 11 2 15 3 13 4 15 4 15 5 17 5 19 5 q + q + q t + q t + 3 q t + q t + 2 q t + 3 q t + q t + 17 6 19 6 19 7 21 7 21 8 23 8 > 3 q t + 2 q t + 3 q t + 3 q t + 2 q t + 3 q t + 23 9 25 9 25 10 27 10 29 11 > 2 q t + 2 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n77

K11n76 K11n78