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

K11n75 K11n77

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X14,5,15,6 X2837 X9,20,10,21 X16,11,17,12 X18,13,19,14 X6,15,7,16 X12,17,13,18 X19,22,20,1 X21,10,22,11

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

t-4 - 3t-3 + 6t-2 - 8t-1 + 9 - 8t + 6t2 - 3t3 + t4

Conway Polynomial:

1 + 5z2 + 8z4 + 5z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{1062, K11n78, ...}

Determinant and Signature:

{45, -4}

Jones Polynomial:

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

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

{K11n78, ...}

A2 (sl(3)) Invariant:

q-28 + q-24 - 3q-22 - 3q-20 - 3q-18 - 2q-16 + 4q-14 + 2q-12 + 6q-10 + 2q-8 + q-6 - 2q-2 - q2

HOMFLY-PT Polynomial:

- 4a2 - 8a2z2 - 5a2z4 - a2z6 + 13a4 + 25a4z2 + 19a4z4 + 7a4z6 + a4z8 - 10a6 - 13a6z2 - 6a6z4 - a6z6 + 2a8 + a8z2

Kauffman Polynomial:

- 4az + 8az3 - 5az5 + az7 + 4a2 - 9a2z2 + 13a2z4 - 9a2z6 + 2a2z8 - 13a3z + 28a3z3 - 18a3z5 + a3z7 + a3z9 + 13a4 - 26a4z2 + 31a4z4 - 24a4z6 + 6a4z8 - 21a5z + 42a5z3 - 31a5z5 + 5a5z7 + a5z9 + 10a6 - 17a6z2 + 16a6z4 - 13a6z6 + 4a6z8 - 14a7z + 26a7z3 - 18a7z5 + 5a7z7 + 2a8 + a8z2 - 2a8z4 + 2a8z6 - 2a9z + 4a9z3 + a10z2

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

{5, -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=-4 is the signature of 1176. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 3                   1 j = 1                 1   j = -1               3 1   j = -3             4 1     j = -5           3 4       j = -7         5 3         j = -9       2 3           j = -11     3 5             j = -13   1 2               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, NonAlternating, 76]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 76]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[14, 5, 15, 6], X[2, 8, 3, 7], > X[9, 20, 10, 21], X[16, 11, 17, 12], X[18, 13, 19, 14], X[6, 15, 7, 16], > X[12, 17, 13, 18], X[19, 22, 20, 1], X[21, 10, 22, 11]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 76]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -8, 4, -2, -5, 11, 6, -9, 7, -3, 8, -6, 9, -7, -10, > 5, -11, 10]
In[5]:=	DTCode[Knot[11, NonAlternating, 76]]
Out[5]=	DTCode[4, 8, 14, 2, -20, 16, 18, 6, 12, -22, -10]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 76]][t]
Out[6]=	-4 3 6 8 2 3 4 9 + t - -- + -- - - - 8 t + 6 t - 3 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 76]][z]
Out[7]=	2 4 6 8 1 + 5 z + 8 z + 5 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 62], Knot[11, NonAlternating, 76], Knot[11, NonAlternating, 78]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 76]], KnotSignature[Knot[11, NonAlternating, 76]]}
Out[9]=	{45, -4}
In[10]:=	J=Jones[Knot[11, NonAlternating, 76]][q]
Out[10]=	-8 4 5 7 8 6 7 4 2 + 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, NonAlternating, 76], Knot[11, NonAlternating, 78]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 76]][q]
Out[12]=	-28 -24 3 3 3 2 4 2 6 2 -6 2 2 q + q - --- - --- - --- - --- + --- + --- + --- + -- + q - -- - q 22 20 18 16 14 12 10 8 2 q q q q q q q q q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 76]][a, z]
Out[13]=	2 4 6 8 2 2 4 2 6 2 8 2 -4 a + 13 a - 10 a + 2 a - 8 a z + 25 a z - 13 a z + a z - 2 4 4 4 6 4 2 6 4 6 6 6 4 8 > 5 a z + 19 a z - 6 a z - a z + 7 a z - a z + a z
In[14]:=	Kauffman[Knot[11, NonAlternating, 76]][a, z]
Out[14]=	2 4 6 8 3 5 7 9 4 a + 13 a + 10 a + 2 a - 4 a z - 13 a z - 21 a z - 14 a z - 2 a z - 2 2 4 2 6 2 8 2 10 2 3 3 3 > 9 a z - 26 a z - 17 a z + a z + a z + 8 a z + 28 a z + 5 3 7 3 9 3 2 4 4 4 6 4 8 4 > 42 a z + 26 a z + 4 a z + 13 a z + 31 a z + 16 a z - 2 a z - 5 3 5 5 5 7 5 2 6 4 6 6 6 > 5 a z - 18 a z - 31 a z - 18 a z - 9 a z - 24 a z - 13 a z + 8 6 7 3 7 5 7 7 7 2 8 4 8 6 8 > 2 a z + a z + a z + 5 a z + 5 a z + 2 a z + 6 a z + 4 a z + 3 9 5 9 > a z + a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 76]], Vassiliev[3][Knot[11, NonAlternating, 76]]}
Out[15]=	{5, -7}
In[16]:=	Kh[Knot[11, NonAlternating, 76]][q, t]
Out[16]=	4 4 1 3 1 2 3 5 2 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 5 3 17 6 15 5 13 5 13 4 11 4 11 3 9 3 9 2 q q q t q t q t q t q t q t q t q t 2 5 3 3 t 3 t t 2 3 3 > ----- + ---- + ---- + -- + --- + -- + q t + q t 7 2 7 5 3 q q q t q t q t q

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n76

K11n75 K11n77