The Alternating Knot 10₅₅

Visit 10₅₅'s page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10₅₅'s page at Knotilus!

Acknowledgement

KnotPlot

PD Presentation: X₁₄₂₅ X₃₈₄₉ X_5,12,6,13 X_15,18,16,19 X_9,16,10,17 X_17,10,18,11 X_13,20,14,1 X_19,14,20,15 X_11,6,12,7 X₇₂₈₃

Gauss Code: {-1, 10, -2, 1, -3, 9, -10, 2, -5, 6, -9, 3, -7, 8, -4, 5, -6, 4, -8, 7}

DT (Dowker-Thistlethwaite) Code: 4 8 12 2 16 6 20 18 10 14

Minimum Braid Representative:

Length is 12, width is 5
Braid index is 5

A Morse Link Presentation:

3D Invariants:

Symmetry Type Unknotting Number 3-Genus Bridge/Super Bridge Index Nakanishi Index

Reversible 2 2 3 / NotAvailable 1

Alexander Polynomial: 5t^-2 - 15t^-1 + 21 - 15t + 5t²

Conway Polynomial: 1 + 5z² + 5z⁴

Other knots with the same Alexander/Conway Polynomial: {...}

Determinant and Signature: {61, -4}

Jones Polynomial: q^-12 - 3q^-11 + 5q^-10 - 8q^-9 + 9q^-8 - 10q^-7 + 10q^-6 - 7q^-5 + 5q^-4 - 2q^-3 + q^-2

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

A2 (sl(3)) Invariant: q^-38 + q^-36 - 2q^-34 - q^-30 - 3q^-28 + q^-26 - q^-24 + q^-22 + q^-20 + 3q^-16 - q^-14 + q^-12 + 2q^-10 - q^-8 + q^-6

HOMFLY-PT Polynomial: a⁴ + 2a⁴z² + a⁴z⁴ + a⁶ + 3a⁶z² + 2a⁶z⁴ + a⁸ + 3a⁸z² + 2a⁸z⁴ - 3a¹⁰ - 3a¹⁰z² + a¹²

Kauffman Polynomial: a⁴ - 2a⁴z² + a⁴z⁴ - 2a⁵z³ + 2a⁵z⁵ - a⁶ + 2a⁶z² - 3a⁶z⁴ + 3a⁶z⁶ + 2a⁷z - 2a⁷z³ - a⁷z⁵ + 3a⁷z⁷ + a⁸ - 3a⁸z² + 5a⁸z⁴ - 4a⁸z⁶ + 3a⁸z⁸ - 4a⁹z + 15a⁹z³ - 16a⁹z⁵ + 5a⁹z⁷ + a⁹z⁹ + 3a¹⁰ - 8a¹⁰z² + 13a¹⁰z⁴ - 15a¹⁰z⁶ + 6a¹⁰z⁸ - 9a¹¹z + 24a¹¹z³ - 23a¹¹z⁵ + 5a¹¹z⁷ + a¹¹z⁹ + a¹² + a¹²z² + a¹²z⁴ - 7a¹²z⁶ + 3a¹²z⁸ - 3a¹³z + 9a¹³z³ - 10a¹³z⁵ + 3a¹³z⁷ + 2a¹⁴z² - 3a¹⁴z⁴ + a¹⁴z⁶

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {5, -10}

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 10₅₅. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

t^rq^j r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0

j = -3 1

j = -5 2 1

j = -7 3

j = -9 4 2

j = -11 6 3

j = -13 4 4

j = -15 5 6

j = -17 3 4

j = -19 2 5

j = -21 1 3

j = -23 2

j = -25 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-34 - 3q^-33 + 10q^-31 - 12q^-30 - 8q^-29 + 31q^-28 - 18q^-27 - 30q^-26 + 55q^-25 - 12q^-24 - 57q^-23 + 68q^-22 + 2q^-21 - 77q^-20 + 67q^-19 + 16q^-18 - 78q^-17 + 52q^-16 + 21q^-15 - 57q^-14 + 29q^-13 + 15q^-12 - 28q^-11 + 13q^-10 + 5q^-9 - 9q^-8 + 5q^-7 + q^-6 - 2q^-5 + q^-4

3 q^-66 - 3q^-65 + 5q^-63 + 5q^-62 - 12q^-61 - 14q^-60 + 19q^-59 + 30q^-58 - 20q^-57 - 56q^-56 + 12q^-55 + 86q^-54 + 11q^-53 - 114q^-52 - 46q^-51 + 126q^-50 + 100q^-49 - 134q^-48 - 145q^-47 + 110q^-46 + 201q^-45 - 85q^-44 - 236q^-43 + 38q^-42 + 275q^-41 - 289q^-39 - 50q^-38 + 302q^-37 + 90q^-36 - 298q^-35 - 125q^-34 + 277q^-33 + 156q^-32 - 250q^-31 - 160q^-30 + 193q^-29 + 169q^-28 - 153q^-27 - 140q^-26 + 92q^-25 + 118q^-24 - 61q^-23 - 73q^-22 + 23q^-21 + 51q^-20 - 20q^-19 - 16q^-18 + 5q^-17 + 10q^-16 - 9q^-15 + 2q^-14 + 4q^-13 + q^-12 - 6q^-11 + 3q^-10 + q^-9 + q^-8 - 2q^-7 + q^-6

4 q^-108 - 3q^-107 + 5q^-105 + 5q^-103 - 19q^-102 - 7q^-101 + 20q^-100 + 11q^-99 + 36q^-98 - 58q^-97 - 53q^-96 + 19q^-95 + 35q^-94 + 149q^-93 - 69q^-92 - 142q^-91 - 77q^-90 - 17q^-89 + 346q^-88 + 57q^-87 - 144q^-86 - 255q^-85 - 283q^-84 + 445q^-83 + 291q^-82 + 116q^-81 - 301q^-80 - 715q^-79 + 248q^-78 + 390q^-77 + 583q^-76 - 16q^-75 - 1045q^-74 - 201q^-73 + 168q^-72 + 1009q^-71 + 527q^-70 - 1093q^-69 - 667q^-68 - 295q^-67 + 1225q^-66 + 1100q^-65 - 905q^-64 - 1005q^-63 - 802q^-62 + 1259q^-61 + 1555q^-60 - 618q^-59 - 1198q^-58 - 1230q^-57 + 1149q^-56 + 1845q^-55 - 269q^-54 - 1221q^-53 - 1537q^-52 + 862q^-51 + 1891q^-50 + 130q^-49 - 985q^-48 - 1626q^-47 + 399q^-46 + 1590q^-45 + 446q^-44 - 514q^-43 - 1379q^-42 - 43q^-41 + 1001q^-40 + 492q^-39 - 42q^-38 - 867q^-37 - 237q^-36 + 421q^-35 + 296q^-34 + 177q^-33 - 377q^-32 - 180q^-31 + 98q^-30 + 78q^-29 + 159q^-28 - 107q^-27 - 66q^-26 + 6q^-25 - 16q^-24 + 76q^-23 - 21q^-22 - 8q^-21 - 24q^-19 + 25q^-18 - 4q^-17 + 3q^-16 + 2q^-15 - 10q^-14 + 6q^-13 - q^-12 + q^-11 + q^-10 - 2q^-9 + q^-8

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]:=
PD[Knot[10, 55]]

Out[2]=
PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[5, 12, 6, 13], X[15, 18, 16, 19], > X[9, 16, 10, 17], X[17, 10, 18, 11], X[13, 20, 14, 1], X[19, 14, 20, 15], > X[11, 6, 12, 7], X[7, 2, 8, 3]]

In[3]:=
GaussCode[Knot[10, 55]]

Out[3]=
GaussCode[-1, 10, -2, 1, -3, 9, -10, 2, -5, 6, -9, 3, -7, 8, -4, 5, -6, 4, -8, > 7]

In[4]:=
DTCode[Knot[10, 55]]

Out[4]=
DTCode[4, 8, 12, 2, 16, 6, 20, 18, 10, 14]

In[5]:=
br = BR[Knot[10, 55]]

Out[5]=
BR[5, {-1, -1, -1, -2, 1, 3, -2, -4, -3, -3, -3, -4}]

In[6]:=
{First[br], Crossings[br]}

Out[6]=
{5, 12}

In[7]:=
BraidIndex[Knot[10, 55]]

Out[7]=
5

In[8]:=
Show[DrawMorseLink[Knot[10, 55]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 55]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

Out[9]=
{Reversible, 2, 2, 3, NotAvailable, 1}

In[10]:=
alex = Alexander[Knot[10, 55]][t]

Out[10]=
5 15 2 21 + -- - -- - 15 t + 5 t 2 t t

In[11]:=
Conway[Knot[10, 55]][z]

Out[11]=
2 4 1 + 5 z + 5 z

In[12]:=
Select[AllKnots[], (alex === Alexander[#][t])&]

Out[12]=
{Knot[10, 55]}

In[13]:=
{KnotDet[Knot[10, 55]], KnotSignature[Knot[10, 55]]}

Out[13]=
{61, -4}

In[14]:=
Jones[Knot[10, 55]][q]

Out[14]=
-12 3 5 8 9 10 10 7 5 2 -2 q - --- + --- - -- + -- - -- + -- - -- + -- - -- + q 11 10 9 8 7 6 5 4 3 q q q q q q q q q

In[15]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]

Out[15]=
{Knot[10, 55]}

In[16]:=
A2Invariant[Knot[10, 55]][q]

Out[16]=
-38 -36 2 -30 3 -26 -24 -22 -20 3 -14 q + q - --- - q - --- + q - q + q + q + --- - q + 34 28 16 q q q -12 2 -8 -6 > q + --- - q + q 10 q

In[17]:=
HOMFLYPT[Knot[10, 55]][a, z]

Out[17]=
4 6 8 10 12 4 2 6 2 8 2 10 2 4 4 a + a + a - 3 a + a + 2 a z + 3 a z + 3 a z - 3 a z + a z + 6 4 8 4 > 2 a z + 2 a z

In[18]:=
Kauffman[Knot[10, 55]][a, z]

Out[18]=
4 6 8 10 12 7 9 11 13 4 2 a - a + a + 3 a + a + 2 a z - 4 a z - 9 a z - 3 a z - 2 a z + 6 2 8 2 10 2 12 2 14 2 5 3 7 3 > 2 a z - 3 a z - 8 a z + a z + 2 a z - 2 a z - 2 a z + 9 3 11 3 13 3 4 4 6 4 8 4 10 4 > 15 a z + 24 a z + 9 a z + a z - 3 a z + 5 a z + 13 a z + 12 4 14 4 5 5 7 5 9 5 11 5 13 5 > a z - 3 a z + 2 a z - a z - 16 a z - 23 a z - 10 a z + 6 6 8 6 10 6 12 6 14 6 7 7 9 7 > 3 a z - 4 a z - 15 a z - 7 a z + a z + 3 a z + 5 a z + 11 7 13 7 8 8 10 8 12 8 9 9 11 9 > 5 a z + 3 a z + 3 a z + 6 a z + 3 a z + a z + a z

In[19]:=
{Vassiliev[2][Knot[10, 55]], Vassiliev[3][Knot[10, 55]]}

Out[19]=
{5, -10}

In[20]:=
Kh[Knot[10, 55]][q, t]

Out[20]=
-5 -3 1 2 1 3 2 5 3 q + q + ------- + ------ + ------ + ------ + ------ + ------ + ------ + 25 10 23 9 21 9 21 8 19 8 19 7 17 7 q t q t q t q t q t q t q t 4 5 6 4 4 6 3 4 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 17 6 15 6 15 5 13 5 13 4 11 4 11 3 9 3 q t q t q t q t q t q t q t q t 2 3 2 > ----- + ----- + ---- 9 2 7 2 5 q t q t q t

In[21]:=
ColouredJones[Knot[10, 55], 2][q]

Out[21]=
-34 3 10 12 8 31 18 30 55 12 57 68 2 q - --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + --- + --- - 33 31 30 29 28 27 26 25 24 23 22 21 q q q q q q q q q q q q 77 67 16 78 52 21 57 29 15 28 13 5 9 > --- + --- + --- - --- + --- + --- - --- + --- + --- - --- + --- + -- - -- + 20 19 18 17 16 15 14 13 12 11 10 9 8 q q q q q q q q q q q q q 5 -6 2 -4 > -- + q - -- + q 7 5 q q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₅₅

10₅₄
10₅₆

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-34 - 3q^-33 + 10q^-31 - 12q^-30 - 8q^-29 + 31q^-28 - 18q^-27 - 30q^-26 + 55q^-25 - 12q^-24 - 57q^-23 + 68q^-22 + 2q^-21 - 77q^-20 + 67q^-19 + 16q^-18 - 78q^-17 + 52q^-16 + 21q^-15 - 57q^-14 + 29q^-13 + 15q^-12 - 28q^-11 + 13q^-10 + 5q^-9 - 9q^-8 + 5q^-7 + q^-6 - 2q^-5 + q^-4
3	q^-66 - 3q^-65 + 5q^-63 + 5q^-62 - 12q^-61 - 14q^-60 + 19q^-59 + 30q^-58 - 20q^-57 - 56q^-56 + 12q^-55 + 86q^-54 + 11q^-53 - 114q^-52 - 46q^-51 + 126q^-50 + 100q^-49 - 134q^-48 - 145q^-47 + 110q^-46 + 201q^-45 - 85q^-44 - 236q^-43 + 38q^-42 + 275q^-41 - 289q^-39 - 50q^-38 + 302q^-37 + 90q^-36 - 298q^-35 - 125q^-34 + 277q^-33 + 156q^-32 - 250q^-31 - 160q^-30 + 193q^-29 + 169q^-28 - 153q^-27 - 140q^-26 + 92q^-25 + 118q^-24 - 61q^-23 - 73q^-22 + 23q^-21 + 51q^-20 - 20q^-19 - 16q^-18 + 5q^-17 + 10q^-16 - 9q^-15 + 2q^-14 + 4q^-13 + q^-12 - 6q^-11 + 3q^-10 + q^-9 + q^-8 - 2q^-7 + q^-6
4	q^-108 - 3q^-107 + 5q^-105 + 5q^-103 - 19q^-102 - 7q^-101 + 20q^-100 + 11q^-99 + 36q^-98 - 58q^-97 - 53q^-96 + 19q^-95 + 35q^-94 + 149q^-93 - 69q^-92 - 142q^-91 - 77q^-90 - 17q^-89 + 346q^-88 + 57q^-87 - 144q^-86 - 255q^-85 - 283q^-84 + 445q^-83 + 291q^-82 + 116q^-81 - 301q^-80 - 715q^-79 + 248q^-78 + 390q^-77 + 583q^-76 - 16q^-75 - 1045q^-74 - 201q^-73 + 168q^-72 + 1009q^-71 + 527q^-70 - 1093q^-69 - 667q^-68 - 295q^-67 + 1225q^-66 + 1100q^-65 - 905q^-64 - 1005q^-63 - 802q^-62 + 1259q^-61 + 1555q^-60 - 618q^-59 - 1198q^-58 - 1230q^-57 + 1149q^-56 + 1845q^-55 - 269q^-54 - 1221q^-53 - 1537q^-52 + 862q^-51 + 1891q^-50 + 130q^-49 - 985q^-48 - 1626q^-47 + 399q^-46 + 1590q^-45 + 446q^-44 - 514q^-43 - 1379q^-42 - 43q^-41 + 1001q^-40 + 492q^-39 - 42q^-38 - 867q^-37 - 237q^-36 + 421q^-35 + 296q^-34 + 177q^-33 - 377q^-32 - 180q^-31 + 98q^-30 + 78q^-29 + 159q^-28 - 107q^-27 - 66q^-26 + 6q^-25 - 16q^-24 + 76q^-23 - 21q^-22 - 8q^-21 - 24q^-19 + 25q^-18 - 4q^-17 + 3q^-16 + 2q^-15 - 10q^-14 + 6q^-13 - q^-12 + q^-11 + q^-10 - 2q^-9 + q^-8

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 55]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[5, 12, 6, 13], X[15, 18, 16, 19], > X[9, 16, 10, 17], X[17, 10, 18, 11], X[13, 20, 14, 1], X[19, 14, 20, 15], > X[11, 6, 12, 7], X[7, 2, 8, 3]]
In[3]:=	GaussCode[Knot[10, 55]]
Out[3]=	GaussCode[-1, 10, -2, 1, -3, 9, -10, 2, -5, 6, -9, 3, -7, 8, -4, 5, -6, 4, -8, > 7]
In[4]:=	DTCode[Knot[10, 55]]
Out[4]=	DTCode[4, 8, 12, 2, 16, 6, 20, 18, 10, 14]
In[5]:=	br = BR[Knot[10, 55]]
Out[5]=	BR[5, {-1, -1, -1, -2, 1, 3, -2, -4, -3, -3, -3, -4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 12}
In[7]:=	BraidIndex[Knot[10, 55]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[10, 55]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 55]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 2, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 55]][t]
Out[10]=	5 15 2 21 + -- - -- - 15 t + 5 t 2 t t
In[11]:=	Conway[Knot[10, 55]][z]
Out[11]=	2 4 1 + 5 z + 5 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 55]}
In[13]:=	{KnotDet[Knot[10, 55]], KnotSignature[Knot[10, 55]]}
Out[13]=	{61, -4}
In[14]:=	Jones[Knot[10, 55]][q]
Out[14]=	-12 3 5 8 9 10 10 7 5 2 -2 q - --- + --- - -- + -- - -- + -- - -- + -- - -- + q 11 10 9 8 7 6 5 4 3 q q q q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 55]}
In[16]:=	A2Invariant[Knot[10, 55]][q]
Out[16]=	-38 -36 2 -30 3 -26 -24 -22 -20 3 -14 q + q - --- - q - --- + q - q + q + q + --- - q + 34 28 16 q q q -12 2 -8 -6 > q + --- - q + q 10 q
In[17]:=	HOMFLYPT[Knot[10, 55]][a, z]
Out[17]=	4 6 8 10 12 4 2 6 2 8 2 10 2 4 4 a + a + a - 3 a + a + 2 a z + 3 a z + 3 a z - 3 a z + a z + 6 4 8 4 > 2 a z + 2 a z
In[18]:=	Kauffman[Knot[10, 55]][a, z]
Out[18]=	4 6 8 10 12 7 9 11 13 4 2 a - a + a + 3 a + a + 2 a z - 4 a z - 9 a z - 3 a z - 2 a z + 6 2 8 2 10 2 12 2 14 2 5 3 7 3 > 2 a z - 3 a z - 8 a z + a z + 2 a z - 2 a z - 2 a z + 9 3 11 3 13 3 4 4 6 4 8 4 10 4 > 15 a z + 24 a z + 9 a z + a z - 3 a z + 5 a z + 13 a z + 12 4 14 4 5 5 7 5 9 5 11 5 13 5 > a z - 3 a z + 2 a z - a z - 16 a z - 23 a z - 10 a z + 6 6 8 6 10 6 12 6 14 6 7 7 9 7 > 3 a z - 4 a z - 15 a z - 7 a z + a z + 3 a z + 5 a z + 11 7 13 7 8 8 10 8 12 8 9 9 11 9 > 5 a z + 3 a z + 3 a z + 6 a z + 3 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 55]], Vassiliev[3][Knot[10, 55]]}
Out[19]=	{5, -10}
In[20]:=	Kh[Knot[10, 55]][q, t]
Out[20]=	-5 -3 1 2 1 3 2 5 3 q + q + ------- + ------ + ------ + ------ + ------ + ------ + ------ + 25 10 23 9 21 9 21 8 19 8 19 7 17 7 q t q t q t q t q t q t q t 4 5 6 4 4 6 3 4 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 17 6 15 6 15 5 13 5 13 4 11 4 11 3 9 3 q t q t q t q t q t q t q t q t 2 3 2 > ----- + ----- + ---- 9 2 7 2 5 q t q t q t
In[21]:=	ColouredJones[Knot[10, 55], 2][q]
Out[21]=	-34 3 10 12 8 31 18 30 55 12 57 68 2 q - --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + --- + --- - 33 31 30 29 28 27 26 25 24 23 22 21 q q q q q q q q q q q q 77 67 16 78 52 21 57 29 15 28 13 5 9 > --- + --- + --- - --- + --- + --- - --- + --- + --- - --- + --- + -- - -- + 20 19 18 17 16 15 14 13 12 11 10 9 8 q q q q q q q q q q q q q 5 -6 2 -4 > -- + q - -- + q 7 5 q q

The Alternating Knot 1055

The Alternating Knot 10₅₅