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

K11a78 K11a80

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X12,6,13,5 X14,7,15,8 X20,10,21,9 X2,11,3,12 X18,13,19,14 X8,15,9,16 X22,17,1,18 X6,20,7,19 X16,21,17,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 12 14 20 2 18 8 22 6 16

Alexander Polynomial:

- t-4 + 6t-3 - 17t-2 + 30t-1 - 35 + 30t - 17t2 + 6t3 - t4

Conway Polynomial:

1 - z4 - 2z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{K11a255, ...}

Determinant and Signature:

{143, -2}

Jones Polynomial:

- q-8 + 4q-7 - 9q-6 + 15q-5 - 20q-4 + 23q-3 - 23q-2 + 20q-1 - 14 + 9q - 4q2 + q3

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

{K11a255, ...}

A2 (sl(3)) Invariant:

- q-24 + q-22 - 2q-18 + 4q-16 - 3q-14 + 2q-12 + q-10 - 3q-8 + 4q-6 - 5q-4 + 4q-2 - q2 + 3q4 - 2q6 + q8

HOMFLY-PT Polynomial:

2 + 3z2 + 3z4 + z6 - 3a2 - 9a2z2 - 10a2z4 - 5a2z6 - a2z8 + 3a4 + 8a4z2 + 7a4z4 + 2a4z6 - a6 - 2a6z2 - a6z4

Kauffman Polynomial:

a-2z2 - 2a-2z4 + a-2z6 - a-1z + 5a-1z3 - 9a-1z5 + 4a-1z7 + 2 - 5z2 + 11z4 - 16z6 + 7z8 - 3az + 10az3 - 9az5 - 6az7 + 6az9 + 3a2 - 19a2z2 + 44a2z4 - 45a2z6 + 14a2z8 + 2a2z10 - 5a3z + 14a3z3 - a3z5 - 20a3z7 + 13a3z9 + 3a4 - 19a4z2 + 47a4z4 - 48a4z6 + 17a4z8 + 2a4z10 - 5a5z + 16a5z3 - 14a5z5 - 2a5z7 + 7a5z9 + a6 - 5a6z2 + 11a6z4 - 16a6z6 + 10a6z8 - 2a7z + 6a7z3 - 12a7z5 + 8a7z7 + a8z2 - 5a8z4 + 4a8z6 - a9z3 + a9z5

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

{0, -1}

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 = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 7                       1 j = 5                     3   j = 3                   6 1   j = 1                 8 3     j = -1               12 6       j = -3             12 9         j = -5           11 11           j = -7         9 12             j = -9       6 11               j = -11     3 9                 j = -13   1 6                   j = -15   3                     j = -17 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, Alternating, 79]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 79]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[14, 7, 15, 8], > X[20, 10, 21, 9], X[2, 11, 3, 12], X[18, 13, 19, 14], X[8, 15, 9, 16], > X[22, 17, 1, 18], X[6, 20, 7, 19], X[16, 21, 17, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 79]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -10, 4, -8, 5, -2, 6, -3, 7, -4, 8, -11, 9, -7, 10, > -5, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 79]]
Out[5]=	DTCode[4, 10, 12, 14, 20, 2, 18, 8, 22, 6, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 79]][t]
Out[6]=	-4 6 17 30 2 3 4 -35 - t + -- - -- + -- + 30 t - 17 t + 6 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 79]][z]
Out[7]=	4 6 8 1 - z - 2 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 79], Knot[11, Alternating, 255]}
In[9]:=	{KnotDet[Knot[11, Alternating, 79]], KnotSignature[Knot[11, Alternating, 79]]}
Out[9]=	{143, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 79]][q]
Out[10]=	-8 4 9 15 20 23 23 20 2 3 -14 - q + -- - -- + -- - -- + -- - -- + -- + 9 q - 4 q + q 7 6 5 4 3 2 q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 79], Knot[11, Alternating, 255]}
In[12]:=	A2Invariant[Knot[11, Alternating, 79]][q]
Out[12]=	-24 -22 2 4 3 2 -10 3 4 5 4 2 4 -q + q - --- + --- - --- + --- + q - -- + -- - -- + -- - q + 3 q - 18 16 14 12 8 6 4 2 q q q q q q q q 6 8 > 2 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 79]][a, z]
Out[13]=	2 4 6 2 2 2 4 2 6 2 4 2 4 2 - 3 a + 3 a - a + 3 z - 9 a z + 8 a z - 2 a z + 3 z - 10 a z + 4 4 6 4 6 2 6 4 6 2 8 > 7 a z - a z + z - 5 a z + 2 a z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 79]][a, z]
Out[14]=	2 2 4 6 z 3 5 7 2 z 2 + 3 a + 3 a + a - - - 3 a z - 5 a z - 5 a z - 2 a z - 5 z + -- - a 2 a 3 2 2 4 2 6 2 8 2 5 z 3 3 3 > 19 a z - 19 a z - 5 a z + a z + ---- + 10 a z + 14 a z + a 4 5 3 7 3 9 3 4 2 z 2 4 4 4 > 16 a z + 6 a z - a z + 11 z - ---- + 44 a z + 47 a z + 2 a 5 6 4 8 4 9 z 5 3 5 5 5 7 5 9 5 > 11 a z - 5 a z - ---- - 9 a z - a z - 14 a z - 12 a z + a z - a 6 7 6 z 2 6 4 6 6 6 8 6 4 z 7 > 16 z + -- - 45 a z - 48 a z - 16 a z + 4 a z + ---- - 6 a z - 2 a a 3 7 5 7 7 7 8 2 8 4 8 6 8 > 20 a z - 2 a z + 8 a z + 7 z + 14 a z + 17 a z + 10 a z + 9 3 9 5 9 2 10 4 10 > 6 a z + 13 a z + 7 a z + 2 a z + 2 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 79]], Vassiliev[3][Knot[11, Alternating, 79]]}
Out[15]=	{0, -1}
In[16]:=	Kh[Knot[11, Alternating, 79]][q, t]
Out[16]=	9 12 1 3 1 6 3 9 6 11 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 9 3 q q t q t q t q t q t q t q t q t 9 12 11 11 12 6 t 2 3 2 > ----- + ----- + ----- + ---- + ---- + --- + 8 q t + 3 q t + 6 q t + 7 3 7 2 5 2 5 3 q q t q t q t q t q t 3 3 5 3 7 4 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a79

K11a78 K11a80