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

K11n126 K11n128

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X18,5,19,6 X7,12,8,13 X9,16,10,17 X2,11,3,12 X13,21,14,20 X15,8,16,9 X22,17,1,18 X6,19,7,20 X21,15,22,14

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

t-3 - 5t-2 + 13t-1 - 17 + 13t - 5t2 + t3

Conway Polynomial:

1 + 2z2 + z4 + z6

Other knots with the same Alexander/Conway Polynomial:

{931, K11n11, K11n22, K11n112, ...}

Determinant and Signature:

{55, -2}

Jones Polynomial:

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

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

{K11n22, ...}

A2 (sl(3)) Invariant:

q-28 - q-24 + q-22 - 3q-20 - q-18 - q-14 + 3q-12 + 3q-8 + q-6 - q-4 + 2q-2 - 1

HOMFLY-PT Polynomial:

- a2z2 - a2z4 + 4a4 + 7a4z2 + 4a4z4 + a4z6 - 4a6 - 5a6z2 - 2a6z4 + a8 + a8z2

Kauffman Polynomial:

az3 - 2a2z2 + 4a2z4 + 2a3z5 + a3z7 + 4a4 - 9a4z2 + 8a4z4 - 3a4z6 + 2a4z8 - 4a5z + 6a5z3 - 6a5z5 + 2a5z7 + a5z9 + 4a6 - 8a6z2 + 8a6z4 - 11a6z6 + 5a6z8 - 6a7z + 16a7z3 - 18a7z5 + 4a7z7 + a7z9 + a8 + a8z2 + a8z4 - 7a8z6 + 3a8z8 - 2a9z + 9a9z3 - 10a9z5 + 3a9z7 + 2a10z2 - 3a10z4 + a10z6

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

{2, -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 11127. 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 r = 1 j = 1                   1 j = -1                 3   j = -3               4 2   j = -5             5 2     j = -7           4 4       j = -9         5 5         j = -11       3 4           j = -13     2 5             j = -15   1 3               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, 127]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 127]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[18, 5, 19, 6], X[7, 12, 8, 13], > X[9, 16, 10, 17], X[2, 11, 3, 12], X[13, 21, 14, 20], X[15, 8, 16, 9], > X[22, 17, 1, 18], X[6, 19, 7, 20], X[21, 15, 22, 14]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 127]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -10, -4, 8, -5, -2, 6, 4, -7, 11, -8, 5, 9, -3, 10, > 7, -11, -9]
In[5]:=	DTCode[Knot[11, NonAlternating, 127]]
Out[5]=	DTCode[4, 10, 18, -12, -16, 2, -20, -8, 22, 6, -14]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 127]][t]
Out[6]=	-3 5 13 2 3 -17 + t - -- + -- + 13 t - 5 t + t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 127]][z]
Out[7]=	2 4 6 1 + 2 z + z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 31], Knot[11, NonAlternating, 11], Knot[11, NonAlternating, 22], > Knot[11, NonAlternating, 112], Knot[11, NonAlternating, 127]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 127]], KnotSignature[Knot[11, NonAlternating, 127]]}
Out[9]=	{55, -2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 127]][q]
Out[10]=	-9 3 5 8 9 9 9 6 4 -1 + 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, 22], Knot[11, NonAlternating, 127]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 127]][q]
Out[12]=	-28 -24 -22 3 -18 -14 3 3 -6 -4 2 -1 + q - q + q - --- - q - q + --- + -- + q - q + -- 20 12 8 2 q q q q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 127]][a, z]
Out[13]=	4 6 8 2 2 4 2 6 2 8 2 2 4 4 4 4 a - 4 a + a - a z + 7 a z - 5 a z + a z - a z + 4 a z - 6 4 4 6 > 2 a z + a z
In[14]:=	Kauffman[Knot[11, NonAlternating, 127]][a, z]
Out[14]=	4 6 8 5 7 9 2 2 4 2 6 2 4 a + 4 a + a - 4 a z - 6 a z - 2 a z - 2 a z - 9 a z - 8 a z + 8 2 10 2 3 5 3 7 3 9 3 2 4 > a z + 2 a z + a z + 6 a z + 16 a z + 9 a z + 4 a z + 4 4 6 4 8 4 10 4 3 5 5 5 7 5 > 8 a z + 8 a z + a z - 3 a z + 2 a z - 6 a z - 18 a z - 9 5 4 6 6 6 8 6 10 6 3 7 5 7 > 10 a z - 3 a z - 11 a z - 7 a z + a z + a z + 2 a z + 7 7 9 7 4 8 6 8 8 8 5 9 7 9 > 4 a z + 3 a z + 2 a z + 5 a z + 3 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 127]], Vassiliev[3][Knot[11, NonAlternating, 127]]}
Out[15]=	{2, -2}
In[16]:=	Kh[Knot[11, NonAlternating, 127]][q, t]
Out[16]=	2 3 1 2 1 3 2 5 3 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 19 8 17 7 15 7 15 6 13 6 13 5 11 5 q q t q t q t q t q t q t q t 4 5 5 4 4 5 2 4 > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + q t 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 K11n127

K11n126 K11n128