10 121

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 2 t^3-11 t^2+27 t-35+27 t^{-1} -11 t^{-2} +2 t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ 2 z^6+z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 115, -2 }
Jones polynomial	[math]\displaystyle{ -q^2+5 q-10+15 q^{-1} -18 q^{-2} +20 q^{-3} -18 q^{-4} +14 q^{-5} -9 q^{-6} +4 q^{-7} - q^{-8} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^4 a^6-z^2 a^6-a^6+z^6 a^4+2 z^4 a^4+3 z^2 a^4+2 a^4+z^6 a^2+z^4 a^2-z^2 a^2-a^2-z^4+1 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^5 a^9-z^3 a^9+4 z^6 a^8-5 z^4 a^8+z^2 a^8+8 z^7 a^7-13 z^5 a^7+8 z^3 a^7-2 z a^7+9 z^8 a^6-14 z^6 a^6+9 z^4 a^6-3 z^2 a^6+a^6+4 z^9 a^5+9 z^7 a^5-28 z^5 a^5+19 z^3 a^5-3 z a^5+19 z^8 a^4-36 z^6 a^4+22 z^4 a^4-7 z^2 a^4+2 a^4+4 z^9 a^3+11 z^7 a^3-30 z^5 a^3+14 z^3 a^3-z a^3+10 z^8 a^2-13 z^6 a^2+3 z^4 a^2-3 z^2 a^2+a^2+10 z^7 a-15 z^5 a+4 z^3 a+5 z^6-5 z^4+1+z^5 a^{-1} }[/math]
The A2 invariant	[math]\displaystyle{ -q^{24}+2 q^{22}-2 q^{20}-2 q^{18}+4 q^{16}-3 q^{14}+3 q^{12}-q^8+3 q^6-4 q^4+4 q^2-1- q^{-2} +3 q^{-4} - q^{-6} }[/math]
The G2 invariant	[math]\displaystyle{ q^{128}-3 q^{126}+7 q^{124}-13 q^{122}+16 q^{120}-16 q^{118}+7 q^{116}+17 q^{114}-48 q^{112}+88 q^{110}-120 q^{108}+119 q^{106}-76 q^{104}-33 q^{102}+190 q^{100}-339 q^{98}+424 q^{96}-367 q^{94}+144 q^{92}+189 q^{90}-524 q^{88}+710 q^{86}-650 q^{84}+336 q^{82}+111 q^{80}-519 q^{78}+707 q^{76}-574 q^{74}+195 q^{72}+258 q^{70}-566 q^{68}+567 q^{66}-274 q^{64}-195 q^{62}+623 q^{60}-813 q^{58}+696 q^{56}-280 q^{54}-274 q^{52}+766 q^{50}-1016 q^{48}+928 q^{46}-540 q^{44}-20 q^{42}+548 q^{40}-851 q^{38}+845 q^{36}-520 q^{34}+30 q^{32}+417 q^{30}-637 q^{28}+517 q^{26}-141 q^{24}-311 q^{22}+622 q^{20}-634 q^{18}+356 q^{16}+94 q^{14}-517 q^{12}+736 q^{10}-680 q^8+389 q^6-3 q^4-331 q^2+497-471 q^{-2} +323 q^{-4} -115 q^{-6} -58 q^{-8} +155 q^{-10} -181 q^{-12} +143 q^{-14} -81 q^{-16} +29 q^{-18} +10 q^{-20} -25 q^{-22} +26 q^{-24} -20 q^{-26} +10 q^{-28} -4 q^{-30} + q^{-32} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_121.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{17}+3 q^{15}-5 q^{13}+5 q^{11}-4 q^9+2 q^7+2 q^5-3 q^3+5 q-5 q^{-1} +4 q^{-3} - q^{-5} }[/math]
2	[math]\displaystyle{ q^{48}-3 q^{46}+q^{44}+10 q^{42}-18 q^{40}-5 q^{38}+44 q^{36}-30 q^{34}-40 q^{32}+71 q^{30}-9 q^{28}-68 q^{26}+53 q^{24}+24 q^{22}-53 q^{20}+4 q^{18}+40 q^{16}-9 q^{14}-45 q^{12}+35 q^{10}+38 q^8-71 q^6+8 q^4+68 q^2-53-21 q^{-2} +53 q^{-4} -14 q^{-6} -21 q^{-8} +15 q^{-10} + q^{-12} -4 q^{-14} + q^{-16} }[/math]
3	[math]\displaystyle{ -q^{93}+3 q^{91}-q^{89}-6 q^{87}+3 q^{85}+17 q^{83}-4 q^{81}-53 q^{79}-q^{77}+115 q^{75}+51 q^{73}-191 q^{71}-179 q^{69}+243 q^{67}+366 q^{65}-204 q^{63}-573 q^{61}+55 q^{59}+731 q^{57}+168 q^{55}-768 q^{53}-409 q^{51}+679 q^{49}+597 q^{47}-501 q^{45}-688 q^{43}+279 q^{41}+681 q^{39}-47 q^{37}-619 q^{35}-151 q^{33}+509 q^{31}+336 q^{29}-384 q^{27}-507 q^{25}+236 q^{23}+655 q^{21}-53 q^{19}-761 q^{17}-159 q^{15}+782 q^{13}+388 q^{11}-694 q^9-577 q^7+507 q^5+671 q^3-262 q-634 q^{-1} +29 q^{-3} +499 q^{-5} +116 q^{-7} -317 q^{-9} -154 q^{-11} +150 q^{-13} +127 q^{-15} -49 q^{-17} -76 q^{-19} +14 q^{-21} +28 q^{-23} + q^{-25} -11 q^{-27} - q^{-29} +4 q^{-31} - q^{-33} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{24}+2 q^{22}-2 q^{20}-2 q^{18}+4 q^{16}-3 q^{14}+3 q^{12}-q^8+3 q^6-4 q^4+4 q^2-1- q^{-2} +3 q^{-4} - q^{-6} }[/math]
1,1	[math]\displaystyle{ q^{68}-6 q^{66}+20 q^{64}-50 q^{62}+111 q^{60}-224 q^{58}+412 q^{56}-712 q^{54}+1155 q^{52}-1734 q^{50}+2410 q^{48}-3106 q^{46}+3677 q^{44}-3940 q^{42}+3770 q^{40}-3072 q^{38}+1810 q^{36}-94 q^{34}-1888 q^{32}+3880 q^{30}-5664 q^{28}+7012 q^{26}-7730 q^{24}+7754 q^{22}-7064 q^{20}+5760 q^{18}-3990 q^{16}+1976 q^{14}+17 q^{12}-1768 q^{10}+3064 q^8-3800 q^6+3997 q^4-3742 q^2+3190-2472 q^{-2} +1767 q^{-4} -1174 q^{-6} +712 q^{-8} -388 q^{-10} +194 q^{-12} -88 q^{-14} +32 q^{-16} -8 q^{-18} + q^{-20} }[/math]
2,0	[math]\displaystyle{ q^{62}-2 q^{60}+5 q^{56}-3 q^{54}-9 q^{52}+22 q^{48}+2 q^{46}-29 q^{44}-2 q^{42}+29 q^{40}+q^{38}-36 q^{36}+4 q^{34}+29 q^{32}-4 q^{30}-23 q^{28}+13 q^{26}+13 q^{24}-18 q^{22}+8 q^{20}+8 q^{18}-14 q^{16}-6 q^{14}+25 q^{12}-q^{10}-33 q^8+9 q^6+34 q^4-10 q^2-30+16 q^{-2} +24 q^{-4} -9 q^{-6} -16 q^{-8} +2 q^{-10} +10 q^{-12} -2 q^{-14} -3 q^{-16} + q^{-18} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{54}-3 q^{52}+q^{50}+8 q^{48}-14 q^{46}+2 q^{44}+26 q^{42}-35 q^{40}+q^{38}+45 q^{36}-50 q^{34}-q^{32}+46 q^{30}-39 q^{28}-7 q^{26}+29 q^{24}-6 q^{22}-12 q^{20}-q^{18}+24 q^{16}-5 q^{14}-35 q^{12}+41 q^{10}+7 q^8-53 q^6+43 q^4+9 q^2-41+29 q^{-2} +5 q^{-4} -19 q^{-6} +11 q^{-8} +2 q^{-10} -4 q^{-12} + q^{-14} }[/math]
1,0,0	[math]\displaystyle{ -q^{31}+2 q^{29}-3 q^{27}+q^{25}-3 q^{23}+4 q^{21}-3 q^{19}+4 q^{17}+q^{13}-q^9+2 q^7-4 q^5+4 q^3-2 q+3 q^{-1} -2 q^{-3} +3 q^{-5} - q^{-7} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{68}-2 q^{66}-2 q^{64}+8 q^{62}-16 q^{58}+7 q^{56}+23 q^{54}-15 q^{52}-25 q^{50}+26 q^{48}+23 q^{46}-41 q^{44}-20 q^{42}+46 q^{40}-q^{38}-48 q^{36}+23 q^{34}+39 q^{32}-34 q^{30}-15 q^{28}+42 q^{26}-7 q^{24}-41 q^{22}+27 q^{20}+35 q^{18}-41 q^{16}-14 q^{14}+51 q^{12}-2 q^{10}-45 q^8+12 q^6+31 q^4-17 q^2-19+18 q^{-2} +12 q^{-4} -12 q^{-6} -3 q^{-8} +9 q^{-10} - q^{-12} -3 q^{-14} + q^{-16} }[/math]
1,0,0,0	[math]\displaystyle{ -q^{38}+2 q^{36}-3 q^{34}-3 q^{28}+4 q^{26}-3 q^{24}+4 q^{22}+q^{20}+q^{18}+q^{16}-2 q^{10}+2 q^8-4 q^6+4 q^4-2 q^2+2+2 q^{-2} -2 q^{-4} +3 q^{-6} - q^{-8} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{54}+3 q^{52}-7 q^{50}+14 q^{48}-24 q^{46}+36 q^{44}-50 q^{42}+61 q^{40}-65 q^{38}+61 q^{36}-50 q^{34}+29 q^{32}-2 q^{30}-31 q^{28}+65 q^{26}-93 q^{24}+116 q^{22}-124 q^{20}+123 q^{18}-108 q^{16}+83 q^{14}-51 q^{12}+17 q^{10}+13 q^8-39 q^6+57 q^4-65 q^2+65-57 q^{-2} +47 q^{-4} -31 q^{-6} +19 q^{-8} -10 q^{-10} +4 q^{-12} - q^{-14} }[/math]
1,0	[math]\displaystyle{ q^{88}-3 q^{84}-3 q^{82}+4 q^{80}+11 q^{78}+q^{76}-19 q^{74}-16 q^{72}+18 q^{70}+38 q^{68}-q^{66}-53 q^{64}-29 q^{62}+47 q^{60}+57 q^{58}-21 q^{56}-71 q^{54}-11 q^{52}+64 q^{50}+35 q^{48}-46 q^{46}-47 q^{44}+26 q^{42}+48 q^{40}-10 q^{38}-47 q^{36}+45 q^{32}+11 q^{30}-42 q^{28}-21 q^{26}+40 q^{24}+35 q^{22}-35 q^{20}-50 q^{18}+23 q^{16}+65 q^{14}+2 q^{12}-68 q^{10}-33 q^8+54 q^6+57 q^4-24 q^2-59-7 q^{-2} +43 q^{-4} +26 q^{-6} -20 q^{-8} -25 q^{-10} + q^{-12} +15 q^{-14} +6 q^{-16} -4 q^{-18} -4 q^{-20} + q^{-24} }[/math]

D4 Invariants.

Weight

Invariant

1,0,0,0

[math]\displaystyle{ q^{74}-3 q^{72}+4 q^{70}-6 q^{68}+12 q^{66}-19 q^{64}+23 q^{62}-28 q^{60}+41 q^{58}-49 q^{56}+48 q^{54}-49 q^{52}+52 q^{50}-45 q^{48}+27 q^{46}-22 q^{44}+7 q^{42}+17 q^{40}-40 q^{38}+49 q^{36}-65 q^{34}+89 q^{32}-91 q^{30}+93 q^{28}-95 q^{26}+96 q^{24}-76 q^{22}+64 q^{20}-55 q^{18}+33 q^{16}-7 q^{14}-7 q^{12}+16 q^{10}-36 q^8+49 q^6-49 q^4+51 q^2-52+48 q^{-2} -35 q^{-4} +31 q^{-6} -25 q^{-8} +16 q^{-10} -8 q^{-12} +6 q^{-14} -4 q^{-16} + q^{-18} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{128}-3 q^{126}+7 q^{124}-13 q^{122}+16 q^{120}-16 q^{118}+7 q^{116}+17 q^{114}-48 q^{112}+88 q^{110}-120 q^{108}+119 q^{106}-76 q^{104}-33 q^{102}+190 q^{100}-339 q^{98}+424 q^{96}-367 q^{94}+144 q^{92}+189 q^{90}-524 q^{88}+710 q^{86}-650 q^{84}+336 q^{82}+111 q^{80}-519 q^{78}+707 q^{76}-574 q^{74}+195 q^{72}+258 q^{70}-566 q^{68}+567 q^{66}-274 q^{64}-195 q^{62}+623 q^{60}-813 q^{58}+696 q^{56}-280 q^{54}-274 q^{52}+766 q^{50}-1016 q^{48}+928 q^{46}-540 q^{44}-20 q^{42}+548 q^{40}-851 q^{38}+845 q^{36}-520 q^{34}+30 q^{32}+417 q^{30}-637 q^{28}+517 q^{26}-141 q^{24}-311 q^{22}+622 q^{20}-634 q^{18}+356 q^{16}+94 q^{14}-517 q^{12}+736 q^{10}-680 q^8+389 q^6-3 q^4-331 q^2+497-471 q^{-2} +323 q^{-4} -115 q^{-6} -58 q^{-8} +155 q^{-10} -181 q^{-12} +143 q^{-14} -81 q^{-16} +29 q^{-18} +10 q^{-20} -25 q^{-22} +26 q^{-24} -20 q^{-26} +10 q^{-28} -4 q^{-30} + q^{-32} }[/math]

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["10 121"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

[math]\displaystyle{ 2 t^3-11 t^2+27 t-35+27 t^{-1} -11 t^{-2} +2 t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 2 z^6+z^4+z^2+1 }[/math]

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

[math]\displaystyle{ \{1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 115, -2 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

[math]\displaystyle{ -q^2+5 q-10+15 q^{-1} -18 q^{-2} +20 q^{-3} -18 q^{-4} +14 q^{-5} -9 q^{-6} +4 q^{-7} - q^{-8} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

[math]\displaystyle{ -z^4 a^6-z^2 a^6-a^6+z^6 a^4+2 z^4 a^4+3 z^2 a^4+2 a^4+z^6 a^2+z^4 a^2-z^2 a^2-a^2-z^4+1 }[/math]

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

[math]\displaystyle{ z^5 a^9-z^3 a^9+4 z^6 a^8-5 z^4 a^8+z^2 a^8+8 z^7 a^7-13 z^5 a^7+8 z^3 a^7-2 z a^7+9 z^8 a^6-14 z^6 a^6+9 z^4 a^6-3 z^2 a^6+a^6+4 z^9 a^5+9 z^7 a^5-28 z^5 a^5+19 z^3 a^5-3 z a^5+19 z^8 a^4-36 z^6 a^4+22 z^4 a^4-7 z^2 a^4+2 a^4+4 z^9 a^3+11 z^7 a^3-30 z^5 a^3+14 z^3 a^3-z a^3+10 z^8 a^2-13 z^6 a^2+3 z^4 a^2-3 z^2 a^2+a^2+10 z^7 a-15 z^5 a+4 z^3 a+5 z^6-5 z^4+1+z^5 a^{-1} }[/math]