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

K11n79 K11n81

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8394 X14,6,15,5 X10,8,11,7 X2,9,3,10 X11,18,12,19 X6,14,7,13 X15,20,16,21 X17,22,18,1 X19,12,20,13 X21,16,22,17

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- t-3 + t-2 + 5t-1 - 9 + 5t + t2 - t3

Conway Polynomial:

1 - 5z4 - z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{15, -2}

Jones Polynomial:

- q-8 + 2q-7 - 3q-6 + 4q-5 - 3q-4 + 3q-3 - 2q-2 + 1 - q + q2

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

{...}

A2 (sl(3)) Invariant:

- q-26 - q-24 + q-22 - q-20 + q-18 + 2q-16 + q-14 + 2q-12 - q-10 - 2q-6 - 2q-4 + q-2 + q2 + q4 + q6

HOMFLY-PT Polynomial:

3 + 4z2 + z4 - 5a2 - 9a2z2 - 6a2z4 - a2z6 + 3a4 + 3a4z2 + a6 + 2a6z2 - a8

Kauffman Polynomial:

3 - 12z2 + 15z4 - 7z6 + z8 + 3az - 10az3 + 14az5 - 7az7 + az9 + 5a2 - 26a2z2 + 36a2z4 - 16a2z6 + 2a2z8 + 5a3z - 13a3z3 + 18a3z5 - 8a3z7 + a3z9 + 3a4 - 15a4z2 + 20a4z4 - 8a4z6 + a4z8 + 3a5z - 6a5z3 + 3a5z5 - a6 + 3a6z2 - 7a6z4 + 3a6z6 + 3a7z - 6a7z3 + a7z7 - a8 + 4a8z2 - 6a8z4 + 2a8z6 + 2a9z - 3a9z3 + a9z5

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

{0, -3}

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 1180. 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 = 5                       1 j = 3                         j = 1                   1 1   j = -1               2 1       j = -3             2 1 1       j = -5           2 2 1         j = -7         2 2             j = -9       2 2 1             j = -11     1 2                 j = -13   1 2                   j = -15   1                     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, NonAlternating, 80]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 80]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[14, 6, 15, 5], X[10, 8, 11, 7], > X[2, 9, 3, 10], X[11, 18, 12, 19], X[6, 14, 7, 13], X[15, 20, 16, 21], > X[17, 22, 18, 1], X[19, 12, 20, 13], X[21, 16, 22, 17]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 80]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -7, 4, -2, 5, -4, -6, 10, 7, -3, -8, 11, -9, 6, -10, > 8, -11, 9]
In[5]:=	DTCode[Knot[11, NonAlternating, 80]]
Out[5]=	DTCode[4, 8, 14, 10, 2, -18, 6, -20, -22, -12, -16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 80]][t]
Out[6]=	-3 -2 5 2 3 -9 - t + t + - + 5 t + t - t t
In[7]:=	Conway[Knot[11, NonAlternating, 80]][z]
Out[7]=	4 6 1 - 5 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 80]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 80]], KnotSignature[Knot[11, NonAlternating, 80]]}
Out[9]=	{15, -2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 80]][q]
Out[10]=	-8 2 3 4 3 3 2 2 1 - q + -- - -- + -- - -- + -- - -- - q + q 7 6 5 4 3 2 q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 80]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 80]][q]
Out[12]=	-26 -24 -22 -20 -18 2 -14 2 -10 2 2 -2 -q - q + q - q + q + --- + q + --- - q - -- - -- + q + 16 12 6 4 q q q q 2 4 6 > q + q + q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 80]][a, z]
Out[13]=	2 4 6 8 2 2 2 4 2 6 2 4 2 4 3 - 5 a + 3 a + a - a + 4 z - 9 a z + 3 a z + 2 a z + z - 6 a z - 2 6 > a z
In[14]:=	Kauffman[Knot[11, NonAlternating, 80]][a, z]
Out[14]=	2 4 6 8 3 5 7 9 2 3 + 5 a + 3 a - a - a + 3 a z + 5 a z + 3 a z + 3 a z + 2 a z - 12 z - 2 2 4 2 6 2 8 2 3 3 3 5 3 > 26 a z - 15 a z + 3 a z + 4 a z - 10 a z - 13 a z - 6 a z - 7 3 9 3 4 2 4 4 4 6 4 8 4 > 6 a z - 3 a z + 15 z + 36 a z + 20 a z - 7 a z - 6 a z + 5 3 5 5 5 9 5 6 2 6 4 6 > 14 a z + 18 a z + 3 a z + a z - 7 z - 16 a z - 8 a z + 6 6 8 6 7 3 7 7 7 8 2 8 4 8 > 3 a z + 2 a z - 7 a z - 8 a z + a z + z + 2 a z + a z + 9 3 9 > a z + a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 80]], Vassiliev[3][Knot[11, NonAlternating, 80]]}
Out[15]=	{0, -3}
In[16]:=	Kh[Knot[11, NonAlternating, 80]][q, t]
Out[16]=	-5 -3 2 1 1 1 2 1 2 2 q + q + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 q t q t q t q t q t q t q t 2 2 1 2 2 2 2 t t 2 > ----- + ----- + ----- + ----- + ----- + ---- + ---- + -- + - + q t + 9 3 7 3 9 2 7 2 5 2 5 3 3 q q t q t q t q t q t q t q t q 3 5 4 > q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n80

K11n79 K11n81