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

K11n83 K11n85

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8394 X5,18,6,19 X7,12,8,13 X2,9,3,10 X11,17,12,16 X13,20,14,21 X15,6,16,7 X17,11,18,10 X19,22,20,1 X21,14,22,15

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- 2t-2 + 9t-1 - 13 + 9t - 2t2

Conway Polynomial:

1 + z2 - 2z4

Other knots with the same Alexander/Conway Polynomial:

{912, ...}

Determinant and Signature:

{35, -2}

Jones Polynomial:

q-9 - 3q-8 + 4q-7 - 5q-6 + 6q-5 - 6q-4 + 5q-3 - 3q-2 + 2q-1

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

{...}

A2 (sl(3)) Invariant:

q-28 - q-26 - q-24 + q-22 - q-20 + q-18 - q-14 - q-10 + 2q-8 + q-6 + 2q-2

HOMFLY-PT Polynomial:

2a2 + 2a2z2 - a4 - a4z2 - a4z4 - a6z2 - a6z4 + a8z2

Kauffman Polynomial:

- 2a2 + 2a2z2 + a3z5 - a4 - a4z2 + 7a4z4 - 4a4z6 + a4z8 - 2a5z + 3a5z3 + 2a5z5 - 3a5z7 + a5z9 - 6a6z2 + 19a6z4 - 16a6z6 + 4a6z8 - 3a7z + 11a7z3 - 10a7z5 + a7z9 - 2a8z2 + 9a8z4 - 11a8z6 + 3a8z8 - a9z + 8a9z3 - 11a9z5 + 3a9z7 + a10z2 - 3a10z4 + a10z6

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

{1, -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 1184. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -1                 2 j = -3               2 1 j = -5             3 1   j = -7           3 2     j = -9         3 3       j = -11       2 3         j = -13     2 3           j = -15   1 2             j = -17   2               j = -19 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, 84]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 84]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[5, 18, 6, 19], X[7, 12, 8, 13], > X[2, 9, 3, 10], X[11, 17, 12, 16], X[13, 20, 14, 21], X[15, 6, 16, 7], > X[17, 11, 18, 10], X[19, 22, 20, 1], X[21, 14, 22, 15]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 84]]
Out[4]=	GaussCode[1, -5, 2, -1, -3, 8, -4, -2, 5, 9, -6, 4, -7, 11, -8, 6, -9, 3, -10, > 7, -11, 10]
In[5]:=	DTCode[Knot[11, NonAlternating, 84]]
Out[5]=	DTCode[4, 8, -18, -12, 2, -16, -20, -6, -10, -22, -14]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 84]][t]
Out[6]=	2 9 2 -13 - -- + - + 9 t - 2 t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 84]][z]
Out[7]=	2 4 1 + z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 12], Knot[11, NonAlternating, 84]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 84]], KnotSignature[Knot[11, NonAlternating, 84]]}
Out[9]=	{35, -2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 84]][q]
Out[10]=	-9 3 4 5 6 6 5 3 2 q - -- + -- - -- + -- - -- + -- - -- + - 8 7 6 5 4 3 2 q q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 84]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 84]][q]
Out[12]=	-28 -26 -24 -22 -20 -18 -14 -10 2 -6 2 q - q - q + q - q + q - q - q + -- + q + -- 8 2 q q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 84]][a, z]
Out[13]=	2 4 2 2 4 2 6 2 8 2 4 4 6 4 2 a - a + 2 a z - a z - a z + a z - a z - a z
In[14]:=	Kauffman[Knot[11, NonAlternating, 84]][a, z]
Out[14]=	2 4 5 7 9 2 2 4 2 6 2 8 2 -2 a - a - 2 a z - 3 a z - a z + 2 a z - a z - 6 a z - 2 a z + 10 2 5 3 7 3 9 3 4 4 6 4 8 4 > a z + 3 a z + 11 a z + 8 a z + 7 a z + 19 a z + 9 a z - 10 4 3 5 5 5 7 5 9 5 4 6 6 6 > 3 a z + a z + 2 a z - 10 a z - 11 a z - 4 a z - 16 a z - 8 6 10 6 5 7 9 7 4 8 6 8 8 8 5 9 > 11 a z + a z - 3 a z + 3 a z + a z + 4 a z + 3 a z + a z + 7 9 > a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 84]], Vassiliev[3][Knot[11, NonAlternating, 84]]}
Out[15]=	{1, -1}
In[16]:=	Kh[Knot[11, NonAlternating, 84]][q, t]
Out[16]=	-3 2 1 2 1 2 2 3 2 q + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + q 19 8 17 7 15 7 15 6 13 6 13 5 11 5 q t q t q t q t q t q t q t 3 3 3 3 2 3 1 2 > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- 11 4 9 4 9 3 7 3 7 2 5 2 5 3 q t q t q t q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n84

K11n83 K11n85