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

K11n78 K11n80

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

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

Conway Polynomial:

1 - 4z2 - 2z4

Other knots with the same Alexander/Conway Polynomial:

{K11n138, ...}

Determinant and Signature:

{15, 2}

Jones Polynomial:

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

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

{K11n138, ...}

A2 (sl(3)) Invariant:

q-10 + q-8 + q-6 + q-4 - q2 - 2q4 - q6 - q8 + q10 + q12 + q14 + q16

HOMFLY-PT Polynomial:

2a-4 + a-4z2 - 2a-2 - 3a-2z2 - a-2z4 - 1 - 3z2 - z4 + 2a2 + a2z2

Kauffman Polynomial:

a-5z + 2a-4 - 12a-4z2 + 16a-4z4 - 7a-4z6 + a-4z8 + 7a-3z - 17a-3z3 + 17a-3z5 - 7a-3z7 + a-3z9 + 2a-2 - 16a-2z2 + 23a-2z4 - 12a-2z6 + 2a-2z8 + 7a-1z - 15a-1z3 + 13a-1z5 - 6a-1z7 + a-1z9 - 1 + 2z2 + 2z4 - 4z6 + z8 + az + 2az3 - 4az5 + az7 - 2a2 + 6a2z2 - 5a2z4 + a2z6

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

{-4, -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 1179. 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 j = 11                 1 j = 9                   j = 7             2 1   j = 5           1       j = 3         1 2       j = 1       2 2         j = -1       1 1         j = -3   1 2             j = -5                   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, 79]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 79]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 14, 6, 15], X[2, 8, 3, 7], > X[9, 21, 10, 20], X[11, 19, 12, 18], X[13, 6, 14, 7], X[15, 22, 16, 1], > X[17, 13, 18, 12], X[19, 11, 20, 10], X[21, 16, 22, 17]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 79]]
Out[4]=	GaussCode[1, -4, 2, -1, -3, 7, 4, -2, -5, 10, -6, 9, -7, 3, -8, 11, -9, 6, -10, > 5, -11, 8]
In[5]:=	DTCode[Knot[11, NonAlternating, 79]]
Out[5]=	DTCode[4, 8, -14, 2, -20, -18, -6, -22, -12, -10, -16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 79]][t]
Out[6]=	2 4 2 -3 - -- + - + 4 t - 2 t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 79]][z]
Out[7]=	2 4 1 - 4 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 79], Knot[11, NonAlternating, 138]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 79]], KnotSignature[Knot[11, NonAlternating, 79]]}
Out[9]=	{15, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 79]][q]
Out[10]=	-3 -2 2 2 3 4 5 -2 + q - 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[11, NonAlternating, 79], Knot[11, NonAlternating, 138]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 79]][q]
Out[12]=	-10 -8 -6 -4 2 4 6 8 10 12 14 16 q + q + q + q - q - 2 q - q - q + q + q + q + q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 79]][a, z]
Out[13]=	2 2 4 2 2 2 2 z 3 z 2 2 4 z -1 + -- - -- + 2 a - 3 z + -- - ---- + a z - z - -- 4 2 4 2 2 a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 79]][a, z]
Out[14]=	2 2 2 2 2 z 7 z 7 z 2 12 z 16 z 2 2 -1 + -- + -- - 2 a + -- + --- + --- + a z + 2 z - ----- - ----- + 6 a z - 4 2 5 3 a 4 2 a a a a a a 3 3 4 4 5 5 17 z 15 z 3 4 16 z 23 z 2 4 17 z 13 z > ----- - ----- + 2 a z + 2 z + ----- + ----- - 5 a z + ----- + ----- - 3 a 4 2 3 a a a a a 6 6 7 7 8 5 6 7 z 12 z 2 6 7 z 6 z 7 8 z > 4 a z - 4 z - ---- - ----- + a z - ---- - ---- + a z + z + -- + 4 2 3 a 4 a a a a 8 9 9 2 z z z > ---- + -- + -- 2 3 a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 79]], Vassiliev[3][Knot[11, NonAlternating, 79]]}
Out[15]=	{-4, -2}
In[16]:=	Kh[Knot[11, NonAlternating, 79]][q, t]
Out[16]=	1 3 1 1 2 1 2 q 3 5 7 2 - + 2 q + q + ----- + ----- + ----- + --- + --- + 2 q t + q t + 2 q t + q 7 4 3 3 3 2 q t t q t q t q t 7 3 11 4 > q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n79

K11n78 K11n80