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

K11n175 K11n177

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X10,3,11,4 X5,16,6,17 X18,7,19,8 X20,10,21,9 X2,11,3,12 X8,13,9,14 X15,4,16,5 X22,17,1,18 X12,20,13,19 X14,21,15,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

t-3 - 6t-2 + 15t-1 - 19 + 15t - 6t2 + t3

Conway Polynomial:

1 + z6

Other knots with the same Alexander/Conway Polynomial:

{K11n125, ...}

Determinant and Signature:

{63, -2}

Jones Polynomial:

q-9 - 3q-8 + 6q-7 - 9q-6 + 10q-5 - 11q-4 + 10q-3 - 7q-2 + 5q-1 - 1

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

{...}

A2 (sl(3)) Invariant:

q-28 - q-24 + 2q-22 - 2q-20 - 3q-14 + q-12 - 2q-10 + 3q-8 + 2q-6 + 3q-2 - 1

HOMFLY-PT Polynomial:

2a2 - a2z4 + 3a4z2 + 3a4z4 + a4z6 - 2a6 - 4a6z2 - 2a6z4 + a8 + a8z2

Kauffman Polynomial:

az3 - 2a2 - 2a2z2 + 5a2z4 + 2a3z + 2a3z7 - 9a4z2 + 18a4z4 - 11a4z6 + 4a4z8 + 2a5z - 2a5z5 - a5z7 + 2a5z9 + 2a6 - 12a6z2 + 23a6z4 - 23a6z6 + 8a6z8 - 2a7z + 8a7z3 - 11a7z5 + 2a7z9 + a8 - 3a8z2 + 7a8z4 - 11a8z6 + 4a8z8 - 2a9z + 7a9z3 - 9a9z5 + 3a9z7 + 2a10z2 - 3a10z4 + a10z6

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

{0, 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 11176. 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                 4   j = -3               4 2   j = -5             6 3     j = -7           5 4       j = -9         5 6         j = -11       4 5           j = -13     2 5             j = -15   1 4               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, 176]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 176]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 3, 11, 4], X[5, 16, 6, 17], X[18, 7, 19, 8], > X[20, 10, 21, 9], X[2, 11, 3, 12], X[8, 13, 9, 14], X[15, 4, 16, 5], > X[22, 17, 1, 18], X[12, 20, 13, 19], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 176]]
Out[4]=	GaussCode[1, -6, 2, 8, -3, -1, 4, -7, 5, -2, 6, -10, 7, -11, -8, 3, 9, -4, 10, > -5, 11, -9]
In[5]:=	DTCode[Knot[11, NonAlternating, 176]]
Out[5]=	DTCode[6, 10, -16, 18, 20, 2, 8, -4, 22, 12, 14]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 176]][t]
Out[6]=	-3 6 15 2 3 -19 + t - -- + -- + 15 t - 6 t + t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 176]][z]
Out[7]=	6 1 + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 125], Knot[11, NonAlternating, 176]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 176]], KnotSignature[Knot[11, NonAlternating, 176]]}
Out[9]=	{63, -2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 176]][q]
Out[10]=	-9 3 6 9 10 11 10 7 5 -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, 176]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 176]][q]
Out[12]=	-28 -24 2 2 3 -12 2 3 2 3 -1 + q - q + --- - --- - --- + q - --- + -- + -- + -- 22 20 14 10 8 6 2 q q q q q q q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 176]][a, z]
Out[13]=	2 6 8 4 2 6 2 8 2 2 4 4 4 6 4 4 6 2 a - 2 a + a + 3 a z - 4 a z + a z - a z + 3 a z - 2 a z + a z
In[14]:=	Kauffman[Knot[11, NonAlternating, 176]][a, z]
Out[14]=	2 6 8 3 5 7 9 2 2 4 2 -2 a + 2 a + a + 2 a z + 2 a z - 2 a z - 2 a z - 2 a z - 9 a z - 6 2 8 2 10 2 3 7 3 9 3 2 4 > 12 a z - 3 a z + 2 a z + a z + 8 a z + 7 a z + 5 a z + 4 4 6 4 8 4 10 4 5 5 7 5 9 5 > 18 a z + 23 a z + 7 a z - 3 a z - 2 a z - 11 a z - 9 a z - 4 6 6 6 8 6 10 6 3 7 5 7 9 7 > 11 a z - 23 a z - 11 a z + a z + 2 a z - a z + 3 a z + 4 8 6 8 8 8 5 9 7 9 > 4 a z + 8 a z + 4 a z + 2 a z + 2 a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 176]], Vassiliev[3][Knot[11, NonAlternating, 176]]}
Out[15]=	{0, 2}
In[16]:=	Kh[Knot[11, NonAlternating, 176]][q, t]
Out[16]=	2 4 1 2 1 4 2 5 4 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 5 5 6 5 4 6 3 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 K11n176

K11n175 K11n177