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

K11n177 K11n179

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X3,11,4,10 X16,6,17,5 X18,7,19,8 X20,10,21,9 X11,5,12,4 X8,14,9,13 X2,16,3,15 X22,17,1,18 X14,19,15,20 X12,22,13,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

2t-3 - 9t-2 + 22t-1 - 29 + 22t - 9t2 + 2t3

Conway Polynomial:

1 + 4z2 + 3z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{95, 2}

Jones Polynomial:

- 2 + 7q - 11q2 + 15q3 - 16q4 + 16q5 - 13q6 + 9q7 - 5q8 + q9

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

{...}

A2 (sl(3)) Invariant:

- 2 + 3q2 - 2q4 + q6 + 4q8 - 2q10 + 4q12 - 2q14 + 2q16 + q18 - 3q20 + 2q22 - 3q24 - q26 + q28

HOMFLY-PT Polynomial:

a-8z2 - 3a-6 - 6a-6z2 - 3a-6z4 + 5a-4 + 12a-4z2 + 8a-4z4 + 2a-4z6 - a-2 - 3a-2z2 - 2a-2z4

Kauffman Polynomial:

- a-10z4 + a-10z6 + 2a-9z + 4a-9z3 - 11a-9z5 + 5a-9z7 - a-8z2 + 11a-8z4 - 20a-8z6 + 8a-8z8 - a-7z + 14a-7z3 - 24a-7z5 + 2a-7z7 + 4a-7z9 + 3a-6 - 15a-6z2 + 35a-6z4 - 44a-6z6 + 17a-6z8 - 5a-5z + 16a-5z3 - 20a-5z5 + 3a-5z7 + 4a-5z9 + 5a-4 - 20a-4z2 + 31a-4z4 - 22a-4z6 + 9a-4z8 - 3a-3z + 9a-3z3 - 7a-3z5 + 6a-3z7 + a-2 - 6a-2z2 + 8a-2z4 + a-2z6 - a-1z + 3a-1z3

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

{4, 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=2 is the signature of 11178. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj 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                 4   j = 15               5 1   j = 13             8 4     j = 11           8 5       j = 9         8 8         j = 7       7 8           j = 5     4 8             j = 3   3 7               j = 1   5                 j = -1 2

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, 178]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 178]]
Out[3]=	PD[X[6, 2, 7, 1], X[3, 11, 4, 10], X[16, 6, 17, 5], X[18, 7, 19, 8], > X[20, 10, 21, 9], X[11, 5, 12, 4], X[8, 14, 9, 13], X[2, 16, 3, 15], > X[22, 17, 1, 18], X[14, 19, 15, 20], X[12, 22, 13, 21]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 178]]
Out[4]=	GaussCode[1, -8, -2, 6, 3, -1, 4, -7, 5, 2, -6, -11, 7, -10, 8, -3, 9, -4, 10, > -5, 11, -9]
In[5]:=	DTCode[Knot[11, NonAlternating, 178]]
Out[5]=	DTCode[6, -10, 16, 18, 20, -4, 8, 2, 22, 14, 12]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 178]][t]
Out[6]=	2 9 22 2 3 -29 + -- - -- + -- + 22 t - 9 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 178]][z]
Out[7]=	2 4 6 1 + 4 z + 3 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 178]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 178]], KnotSignature[Knot[11, NonAlternating, 178]]}
Out[9]=	{95, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 178]][q]
Out[10]=	2 3 4 5 6 7 8 9 -2 + 7 q - 11 q + 15 q - 16 q + 16 q - 13 q + 9 q - 5 q + q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 178]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 178]][q]
Out[12]=	2 4 6 8 10 12 14 16 18 20 -2 + 3 q - 2 q + q + 4 q - 2 q + 4 q - 2 q + 2 q + q - 3 q + 22 24 26 28 > 2 q - 3 q - q + q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 178]][a, z]
Out[13]=	2 2 2 2 4 4 4 6 -3 5 -2 z 6 z 12 z 3 z 3 z 8 z 2 z 2 z -- + -- - a + -- - ---- + ----- - ---- - ---- + ---- - ---- + ---- 6 4 8 6 4 2 6 4 2 4 a a a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 178]][a, z]
Out[14]=	2 2 2 2 3 3 5 -2 2 z z 5 z 3 z z z 15 z 20 z 6 z 4 z -- + -- + a + --- - -- - --- - --- - - - -- - ----- - ----- - ---- + ---- + 6 4 9 7 5 3 a 8 6 4 2 9 a a a a a a a a a a a 3 3 3 3 4 4 4 4 4 5 14 z 16 z 9 z 3 z z 11 z 35 z 31 z 8 z 11 z > ----- + ----- + ---- + ---- - --- + ----- + ----- + ----- + ---- - ----- - 7 5 3 a 10 8 6 4 2 9 a a a a a a a a a 5 5 5 6 6 6 6 6 7 7 24 z 20 z 7 z z 20 z 44 z 22 z z 5 z 2 z > ----- - ----- - ---- + --- - ----- - ----- - ----- + -- + ---- + ---- + 7 5 3 10 8 6 4 2 9 7 a a a a a a a a a a 7 7 8 8 8 9 9 3 z 6 z 8 z 17 z 9 z 4 z 4 z > ---- + ---- + ---- + ----- + ---- + ---- + ---- 5 3 8 6 4 7 5 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 178]], Vassiliev[3][Knot[11, NonAlternating, 178]]}
Out[15]=	{4, 7}
In[16]:=	Kh[Knot[11, NonAlternating, 178]][q, t]
Out[16]=	3 2 3 5 5 2 7 2 7 3 9 3 5 q + 3 q + --- + 7 q t + 4 q t + 8 q t + 7 q t + 8 q t + 8 q t + q t 9 4 11 4 11 5 13 5 13 6 15 6 15 7 > 8 q t + 8 q t + 5 q t + 8 q t + 4 q t + 5 q t + q t + 17 7 19 8 > 4 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n178

K11n177 K11n179